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Neumann, 1954) A group G has boundedly finite conjugacy classes if and only if its commutator subgroup G is finite D  E!E!E!   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Then the commutator subgroup G of G is finited  e wG! E!eG!   A$??$ArabaRossapiccolaPicture 5ArabaRossapiccola0 +T3f3fF>___PPT10..~P'+JD' = @B Di' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<* w%(D' =-g6B fade*<3<* wD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* wD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<* wD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<* x%(D' =-g6B fade*<3<* xD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* xD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<* xD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<* D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<* +8+0+ 0 +"F 0$(  $ $  xֶ0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Yrdrs/downrev.xmlDQK0Co.ղuF:au|mjsSu>;|`:q$[ nG kf deRp&smO."B稠 ϥUCı;Xg0j"t.I&`V U_o֏xZJl7D! 3e/E+Hl6qx>Vor _ ?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Yrdrs/downrev.xmlPKz<$0 g Theorem 3 (R. Baer, 1952) Let G be a group in which the term Zi(G) of the upper central series has finite index for some positive integer i. Then the (i+1)-th term i+1(G) of the lower central series of G is finite  E!$E!M!fE!M!9E!E"@ $ A$??$ArabaRossapiccolaPicture 5ArabaRossapiccola0 +T3f3f___PPT10..~0o+JD' = @B D?' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*$D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*$D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*$+8+0+$0 +" @((  ( (  xz0e0e?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDN1M|fL.KdaX`vi+,oo^[Ӊ9ZV0% K[w3> k, yX-k{>GM}./2G']evxp4I`6 _ovOv?ߔ  \$KEjt+B#style.visibility<*(%(D' =-g6B fade*<3<*(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*(D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|%(D' =-g6B fade*<3<*(|D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(|D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*(|D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(|%(D' =-g6B fade*<3<*(|D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(|D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*(|+8+0+(0 +"t P,(  , ,  x?0e0e?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Z<drs/downrev.xmlDQO0M5M:$˜BP#̄m.m';' ƣqiuÕ.2)jg>P *C2)}YA?qN 1Jj7O4p|MM(fiNs6R7D!|sWZ|2N N@NAPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Z<drs/downrev.xmlPKz<$0 g Corollary A group G is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent B  E!E!E  , A$??$ArabaRossapiccolaPicture 3ArabaRossapiccola0 +T3f3f  ___PPT10 ..~0o+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =-g6B fade*<3<*, D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*, D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*, D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =-g6B fade*<3<*, D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*, D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*, +8+0+,0 +"$ `0(  0 0  xc0e0e?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDN@E$䑸)TBPBbuyΛffN\ޜXugr`+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*0D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*0-%(D' =-g6B fade*<3<*0-D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0-D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*0-+8+0+00 +" >6p4(  4 4  xR0e0e?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!5qOdrs/downrev.xmlDN0DHHܨCKiu*KM:M=8fFo{rr Q^=\ |@[&G\̱oC#} BJ_dЏ@uC!M/Yv- vZl~l}|UYɺEY\݂8_ԓVp3O@􁜂l)PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!5qOdrs/downrev.xmlPKz<$0 g BTheorem 5 (M. De Falco  F. de Giovanni  C. Musella  Y.P. Sysak, 2009) A group G is finite over its hypercentre if and only if it contains a finite normal subgroup N such that G/N is hypercentral f  E!G!@G!E!E  4 A$??$ArabaRossapiccolaPicture 3ArabaRossapiccola0 +T3f3f___PPT10..~0o+JD' = @B D?' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*4 %(D' =-g6B fade*<3<*4 D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4 D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*4 D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*4 J%(D' =-g6B fade*<3<*4 JD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4 JD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*4 JD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*4K%(D' =-g6B fade*<3<*4KD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4KD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*4K+8+0+40 +"D$ 8(  8 8  x`<0e0e?Rectangle 2" PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!,Dւldrs/shapexml.xml\mo8~_O]JJiAMWmޭV Nɑ9P_]i.9̇(N23LHb2*i)%\rgtrIIX W׿vvȺO'ZF# &5fwSNak};Iҹkǰbgɬ[\5sǘ97;ʳJGZ 'N$+9w7;Wr}]eȵ~Z+c;&TTd?08Xu<(/6"˕CE"Vp~el_'2%2%pnqv&ME ]’kakog;p8jCo9*Z̜sc9S^CkhUv["T?TjaVFY~3~[fnϥ'w.=e Z2ݗF7$>e'{0/wm.hֹܽ|[eY<@Upl} ++*8vaꢂy\z Lu)ۻ';`:/xlg!<}JA*i`CS.ɥ0e  d5FJ~62]9NWmk0+B_G$6ԁ6Fh8βk%Mw FV*t'twzGҶ)ǔb3ǥP<6^XXυ*-9_HWz&LWɶi)ήkiȔNz>mxQpΣ F\ZKBhx j!œ0'~S3 ;0Za6m!X^|Q6%&;$uJ?=ݠJRGb&+5J6FG2,pkSu^x쀐E|nNZ\f+V%xTB"> snn& !Tn)-7$qp"Ǧ]'{֕LIBE*0 A|p`[7 L!cP9WhVyKI΋dKyn}3\M&[A(|ЊqHx-jp~Kaaΰ {1ű.nS9v p픨J/KdMk@˧!2e笞w玒d#&G OLٖ^x8i# e?9.d7cZ_`` 5tBIg/cz69kOXn=HH7^*?UN*HYDӎ6ČTav?PK!oPdrs/downrev.xmlDN@M|͘x'[A~,FhVJ O <9'7wr඗ έP͞o& |@X[&G0]^L1ն56>EeM*K2{!:!FWHpS~C -Kʿ7?FA3^==.lw]n"P<^apbrܹJr _ gPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!,Dւl*drs/shapexml.xmlPK-!oP drs/downrev.xmlPK z<$0 g pThe properties C and C" A group G has the property C if the set {X | X d" G} is finite A group G has the property C" if the set {X | X d" G, X infinite} is finite  E!E!G$!E!E!G$!M! E!E!E!G$!E!E!E!E!E!E!E!G$!O$!E!E!E!E!E!E!E!E!E  8 A$??$ArabaRossapiccolaPicture 3ArabaRossapiccola0 +T3f3f___PPT10..~0o+JD' = @B D?' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*8D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*8c%(D' =-g6B fade*<3<*8cD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*8cD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*8cD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*8d%(D' =-g6B fade*<3<*8dD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*8dD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*8d+8+0+80 +"# }u<](  <W <  x`C0e0e?Rectangle 2"  PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!YzNgdrs/shapexml.xml]mO8~_OlKi TUAn%U~\KsM¯vBu+R'g]pV"͕,;+:ep*;EZ2嗼 F΄ZRa,5v-縕[ͬY2iqL 1O%kZU_" uOJ+ x@ TFUrɮXai5'BVQ,ŘR#7qU] ΢3ҟN>1նkukh\uI`L69DDBSkԁ8U "s~?kr";B2'yzX{:7bm!!F4@#(TzhB,"50PHaؚQ}.,Ҫ. cpHcH шdug.&2LmdD@iF=Z@=Zp`QFZ3K K@lwn ]]idŒY?IawL~6ѻ~yůC09&`K5 -0|P<6m8\egɶY K k*Pv\ !~ʊ>uq&~f XHYglb6]8A-g˂xF9C+mYAͪ,uIE Y /.2J iխ"mI4b1Vk]_>WkŴeZ5!<ޜ%SFujԂsi(C!4gg7(uA-DdM$D4Z?Ą{ ǬEgԣU_A.'yk36#1$4PV/Sj oA_ |d*E14Ev} 4FiO?|qq= >v|?"xU7}?d'Vsdr Zk4 YbՂq-$[_ef5I/~uq|p` @2Tۊ0ײunP/$ McY#UɎs)}*}(409`` 73( gerv1~5(XRVkGb3vQbfQr>Lr-g5&#Rع9G*BԞCIg{0y I5իxR8GvTI. 4i}11Y*a:7ZQ{#ߏFq\ M~&J@ 1 muP78w<-wx8ZQpzGF}!wls(jN,g$NOXp+ =0߷yCǒ8eύF?!YyQ[c%z]V:x"6:#LXnE[! "R(,%ҚjEpPlB%a 0GUd17SHM- *3N}iXb-B#*6k=۲:M!FWJẑ$yj*jyYqm~~{Z2uv~A wҏWV a2 Վ|`m49PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!YzNg*drs/shapexml.xmlPK-!Y drs/downrev.xmlPK ;<$0 g O9Tarski groups (i.e. infinite simple groups whose proper non-trivial subgroups have prime order) have obviously the property C A group G is locally graded if every finitely generated non-trivial subgroup of G contains a proper subgroup of finite index All locally (soluble-by-finite) groups are locally graded : |G!G!E!E!G!G!FG!G!fG!G!E!E : < A$??$ArabaRossapiccolaPicture 3ArabaRossapiccola0 +T3f3f___PPT10..~0o+JD' = @B D?' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*<D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*<D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*<8%(D' =-g6B fade*<3<*<8D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<8D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*<8+8+0+<0 +"+ @(  @ @  x0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!6drs/downrev.xmlDN@M|͘x'[RYAC[6#w^{ӊ#8`8H@-nlxIAFu)|vy1L1JD *LPd0 \G6v{ r)M+od$ 66>Ѳ+6 iݭø#9 Q2ؿ?}wm)PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!6drs/downrev.xmlPKs'g/ <$0 g 2Theorem 6 (F. de Giovanni  D.J.S. Robinson, 2005) Let G be a locally graded group with the property C . Then the commutator subgroup G of G is finite\ P4G!2E!G$!2E!  @ A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..@h+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@3%(D' =-g6B fade*<3<*@3D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@3D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*@3D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@4%(D' =-g6B fade*<3<*@4D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@4D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*@4+8+0+@0 +"z   D (  D  D  xa0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!.drs/downrev.xmlDQO0M5M:A2)F/V}<9'7egrF0F1J|πFbk tp3\ڋҹE9 Cr}UF?btK-%3ؘPcGStm:EikuL{ ԇq/WR@&c`5r>m4>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!.drs/downrev.xmlPKeN<$0 g The locally dihedral 2-group is a C"-group with infinite commutator subgroup Let G be a Cernikov group, and let J be its finite residual (i.e. the largest divisible abelian subgroup of G). We say that G is irreducible if [J,G]`"{1} and J has no infinite proper K-invariant subgroups for CG(J)<Kd"Gt "E!G$!M!)E!E!G!G!G!G!G!G!G!G!G! G!G!G! G!G!G!G!G!O!G! t D A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f___PPT10..g+JD' = @B D?' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*DM%(D' =-g6B fade*<3<*DMD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DMD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*DMD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*DN%(D' =-g6B fade*<3<*DND' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DND' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*DND"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Dt%(D' =-g6B fade*<3<*DtD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DtD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*Dt+8+0+D0 +" ,$H (  H H  xa0e0e?Rectangle 3"  PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!w@~sNdrs/shapexml.xmlZn6}/ ZdȗFE n`)1%*V-*I9v|K t, Q9<y.?,҉CzI 4=ܞ(цR!]sM?~2`Ѓ|HFCs1I\/*cM\fP6fX"S4({-'$ѐ.%eXP)E X ]*þEHg@" [9ǚfRyYcp ntPωYr&5!mv{;8/ v{nlU^Gl إ9,Nճ*_n@huNu]zY]cOv8jIBɳb`S~0s݆j5ڝ 3Ev#S!fRC1h],gf,yh^r+S9 Xkz3IRmfN8l6TzaݶOYѺwFH$" l#'>4l6})A7~gcqα 1maEٍ!7`֤!N:'!Kf!kګs|l y4[9xnҕt9]r*B\u^2z|I!5#L$kUi~ef!wQciiV9jY_~H,s,IVG5U*pAc7V卞U~Hѱ ղs2 U+ \{tTeW5vlTڲw_8dwޫ lfYʁUw_WT//S\bڟQo  ud]FsRI5X1Y00m_;۷ժ-Ł|gxS.8;̙#W0 QJTH jR/~n)q/)28J4RRTiQc$Job=*Z20aQ@#V3p\ ѽku˹KwC"*ܕR6oH" vvIK@tu2MۮxeƋ3r>_RCS ~ڲ/TcZByJ𬼇JHvzM +:C^-/# nี1S>mPmb͂:AؐFk pB}p+|smsǴi*G3$!x85XxqH`(G%b+b3PI{3OJ^ W+gX$,Ԯ Rht\]!޺r#s0dr$yKI΋-dc7Z}V~VMJ$mۺV 3qI1,R,7k: WE[,y_3u-?c='(kY,.F+:޼t1^!hkWVo| 3,xyVBƉPM3ϙe5]{n7eL2Y11k0Q*HfBIg/c޽8qx]%3ΏPK!]drs/downrev.xmlD]O0MkteR ?u}!kuo^/ղk<.lt%?%G-5\rq7TΨ?v) R]QB72Е*!ڊKW-Gь+ltxuMtmYM|ɾwaX4nVR@G/`62C ~6_PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!w@~sN*drs/shapexml.xmlPK-!] drs/downrev.xmlPK e'g <$0 g RTheorem 7 (F. de Giovanni  D.J.S. Robinson, 2005) Let G be a locally graded group with the property C". Then either G is finite or G is an irreducible Cernikov group ~ 4G!2E!G$!O$!AE!E!  H A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..g+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H3%(D' =-g6B fade*<3<*H3D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*H3D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*H3D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H4%(D' =-g6B fade*<3<*H4D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*H4D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*H4+8+0+H0 +"R L(  L L  xa0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!;drs/downrev.xmlDQO0M5M:%mK{m|Ǔs|EoZq$g< dK[)*^ :K `10dtq%"Ć ]&e(k2#9F_IᦕI2jhYS5 }bYѦ:Lm?`:^rR޴q@o-& Eh rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!;drs/downrev.xmlPKF<$0 g Recall that a group G is called metahamiltonian if every non-abelian subgroup of G is normal It was proved by G.M. Romalis and N.F. Sesekin that any locally graded metahamiltonian group has finite commutator subgroup G!G! G!G!"G!G! G!E!G!G!fG!  L A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..n}+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L]%(D' =-g6B fade*<3<*L]D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*L]D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*L]D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L^%(D' =-g6B fade*<3<*L^D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*L^D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*L^+8+0+L0 +"(+ P(  P P  x`a0e0e?Rectangle 3"o i PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!wzZdrs/shapexml.xml]O6~--m خPE#rSfMq~{P2>D8c_i.L x!gRE$V gj"LzOy^VE?1y(LE%˥³iSapTF|(Mn#gJ=gPx2NLxH$;2E@,03J#SٯEi _Zg?fRL {m8W*jeg-KEyۢ-L?sC TwBhYM{vwpEz%2":3zipeK0jG6-K s.t)^BdȥK  dՐIRyB|g3轹LO*5ZZEWGnj*H@h&rz'ƣ[ƿ%ŵsWLsX6Z 5\՝غy4 {v 9}bYlRO˴t2Ku$.$Vݯӌ絈N]X>:,M4iO-4ЅD[6w 4;5b裶olcV?"C|&:-W6xb>aNe#w\jZ2I,sO ^wφ:˦|knD 8Q8I|Q"K≽iasi˫R3hDFT r׃ :^dZ@0,Dju6e}꠰z7݉h#{ m&M$s]0ll3j-5;]Z[R%tu&Zp`JBHhiF,[N=bcHVиVMO1+MaDB\ !`B;*vmgN(Λvp9+5^ ,W(Ȕ$L,y j^e"*ȅؙNa7"@8[ a" +6Vܳ!E5`tjEa8Z31R@Et2j+p^jeh:-oY;vR3k qŰcm؁ҺVipػŬ{M֨rq&s)p1fɺ:;[*s__Q^4q[|ΎK>ld\_NdcTZQ3k6XiP9~wɒ|Mb nLL ҄_PRI>TA 僚R51 jBRt0TqsuuϦhPK!<`drs/downrev.xmlDQK0Co.ݤn*a뵹kMRu>;|`:q`Qm픶 [p5"YEpق vˇ]lDPBc_H ٦nAotLpIHCڦz.[vn*+wi>Cx}Aַ "w?._ԓB|bxZm)D/&SPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!wzZ*drs/shapexml.xmlPK-!<`V drs/downrev.xmlPK^ j9 <$0 g vIn fact, Theorem 6 can be proved also if the condition C is imposed only to non-normal subgroups Theorem 8 (F. De Mari  F. de Giovanni, 2006) Let G be a locally graded group with finitely many derived subgroups of non-normal subgroups. Then the commutator subgroup G of G is finite A similar remark holds also for the property C" R Z8G!E!YG!G!-G!G!O!G! R P A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3fF>___PPT10..@"B#+JD' = @B Di' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Pc%(D' =-g6B fade*<3<*PcD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PcD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*PcD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Pd%(D' =-g6B fade*<3<*PdD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PdD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*PdD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-g6B fade*<3<*PD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*PD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P R%(D' =-g6B fade*<3<*P RD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*P RD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*P R+8+0+P0 +"^+   T (  T  T  xa0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!M0. drs/downrev.xmlDn0xkz+NH D[$ۑBxZ=ь7[ 'vv`SY՚x@@FQg #\b>Ql sBhBs)}հ&?=btT;y$S5񡡞W WݷF1+V_ׇtO|xc.æ+Q !˒i bv)];m)PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!M0. drs/downrev.xmlPK  <$0 g The properties K and K" A group G has the property K if for each element x of G the set {[x,H] | H d" G} is finite A group G has the property K" if for each element x of G the set {[x,H] | H d" G, H infinite} is finiteO E!E!G$!E!E!G$!M!E!G!G!G!E!G!G!G!G!G!>G!G!G!!G!G!G!G!G!G!E!O!G!G!G!G!>G!G!G!G!G!G!G!G! O T A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f___PPT10..!+JD' = @B D?' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-g6B fade*<3<*TD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*TD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-g6B fade*<3<*TD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*TD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*TO%(D' =-g6B fade*<3<*TOD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*TOD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*TO+8+0+T0 +"$+ X(  X X  x`a0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!>Qdrs/downrev.xmlDQO0M%M !Q1!a\6Yo6>9NŪ78e0A\ZpxYck\jy}L3锇JD !tɠڎ8v ]%sVAl8>ѦdOMaפwnz"PǩE_ԫV0Kt rtޣDhMA.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!>Qdrs/downrev.xmlPK'g <$0 g As the commutator subgroup of any FC-group is locally finite, it is easy to prove that all FC-groups have the property K On the other hand, also Tarski groups have the property K "G!G!KG!G!G!E!GG!E!G!  X A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..phN+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*XD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*XD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*XD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*X+8+0+X0 +"$+ \(  \ \  x0e0e?Rectangle 3" Ò PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!X&Q' Pdrs/shapexml.xml\o#'~ZIا$RړȊScwx]vI+\/a, N>nlҩcЦHƩ?/(цeR1~I1AaGŘ.)F2Z%ĊݡO BЕڔKOwKbm?ݖcWV-/Œ\.d|OvL.rT}ծlMTV''l$n(+C'26[C" ڝKI^?KZ123Vnd<2T!(S d|%.j?g#fLDuL % ґ fb^DP\o?3UBͥ/YLӺ$fno A+oU|r@FߑrB6>"#naQVRYS'as>\ ˛))ʠk RŒ,|=6XΆ@~bj5eZ.:SZ1M^.,i;2 At#ZgU6y… K2TZ;>YgcYul?p?&P iʐ1q*xXi26\͗Ydkub\mq#pYEj%p+A,?pDvt^t!x%k(*x%F.K \Rk0W&c[ u ) 0MaϲkO~lMή`q{:{z qB>H ˄*r!OxT J_Tzȟ_?_#./i-%jgjui32hn"0ct8~(ªJ;QӠPhʚ `jw{*R Ղ_O>Dkݼ~(ƺ]M~Ւ4ǫ\ϤGH*$%Tt݁#):푑 5(N=XsyT]k84eCWt;c8̘ Κ:'Yyf7r1 MU$'* 4#Fwx[vYpy+1f]]2)f_ֹNl0? n xp_+ g?xa 7&a>ZU|ۺaPK!$drs/downrev.xmlDj1E C8Bj:5؊BA8Ns$TcfofMABF x$HFֺpWc`ΣMn`>M1t9Ev 2]QB70؝Uc%Z$\aC-+*>_J@7ڽm_8_f+QpBM_Hd8I׷'+ Eh |PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!X&Q' P*drs/shapexml.xmlPK-!$ drs/downrev.xmlPK sr <$0 g BTheorem 9 (M. De Falco  F. de Giovanni  C. Musella, 2010) A group G is an FC-group if and only if it is locally (soluble-by-finite) and has the property K  =G!`G!E!E!E!E!E!  \ A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\<%(D' =-g6B fade*<3<*\<D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*\<D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*\<D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\=%(D' =-g6B fade*<3<*\=D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*\=D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*\=+8+0+\0 +",  `(  `y `  x@0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!6Sdrs/downrev.xmlDQO0M5M: !Q0!aܭm.mm|Ǔs|UoZq& qiuÕc02)f>9 )*CR)}YA?qN 1Jj7|L4p|MMWmty47iJ9@}gTnW޵p<8] @NAPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!6Sdrs/downrev.xmlPK ? <$0 g Theorem 10 (M. De Falco  F. de Giovanni  C. Musella, 2010) A soluble-by-finite group G has the property K" if and only if it is either an FC-group or a finite extension of a group of type p" for some prime number p >G!G!-G!E!O!TG!O!G!O!  ` A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..p+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*`>%(D' =-g6B fade*<3<*`>D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*`>D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*`>D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*`?%(D' =-g6B fade*<3<*`?D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*`?D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*`?+8+0+`0 +"%,   0d (  d d  x`@0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!:drs/downrev.xmlDN1M|fL.+!Q01?lM[`y{/|'tF~/&2O0E5pa͔Refçm(DBBJ5mno+rtpAćZ^GЎo$[fv.q翿 ]׼C}*bqٹJmv/FS_PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!:drs/downrev.xmlPKT <$0 g  We shall say that a group G has the property N if for each subgroup X of G the set {[X,H] | H d" G} is finite Theorem 11 (M. De Falco - F. de Giovanni  C. Musella, 2010) Let G be a soluble group with the property N . Then the commutator subgroup G of G is finite  G!G!G!E!E!G!G!G!G! G!G!G!!G!G!G!G!>G!,G!E!3G!G!E!  d A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3fph___PPT10H..09*+JD' = @B D' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*dT%(D' =-g6B fade*<3<*dTD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dTD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*dTD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*dTp%(D' =-g6B fade*<3<*dTpD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dTpD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*dTpD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*dq%(D' =-g6B fade*<3<*dqD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dqD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*dqD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-g6B fade*<3<*dD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*dD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-g6B fade*<3<*dD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*d+8+0+d0 +"!,   @h (  h  h  x0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!zYdrs/downrev.xmlDJ@E0<4 iTXAh㦻g5fބM./r.gM/|gY$A\[qzAL ac퉷t܅FD! nɠ؁8v ]#S^%I* vZhR6 lպ3۫RWj"i7+QVf<qx>Nor _ ?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!zYdrs/downrev.xmlPKF9"<$0 g Let G be a group and let X be a subgroup of G. X is said to be inert in G if the index |X:X Xg| is finite for each element g of G X is said to be strongly inert in G if the index |X,Xg:X| is finite for each element g of G  P PG!G!G!G!G!G!G!G!G!G! G!G!!!!&!!!!!!!!!!!!&!!!!!!G!G!  h A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3fF>___PPT10..+JD' = @B Di' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*h/%(D' =-g6B fade*<3<*h/D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*h/D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*h/D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*h0%(D' =-g6B fade*<3<*h0D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*h0D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*h0D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-g6B fade*<3<*hD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*hD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*hD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-g6B fade*<3<*hD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*hD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*h+8+0+h0 +", Pl(  l} l  x 0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!lMdrs/downrev.xmlDN1M|fLVWBݴB/OwMiŁ}hE(lK[!W!:'0M)h|J$ 9!1vPm)JjO7*5jxYs6]YI,ͦxy/ADv^25݁?>|"{l)PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!lMdrs/downrev.xmlPKz9 <$0 g A group G is called inertial if all its subgroups are inert Similarly, G is strongly inertial if every subgroup of G is strongly inert   E!E! E!E!,E!E!E!E!E!E!E!!A  l A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ..+p3g+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*l<%(D' =-g6B fade*<3<*l<D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*l<D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*l<D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*l=%(D' =-g6B fade*<3<*l=D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*l=D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*l=+8+0+l0 +" P2   `p (  p p  x a0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Pdrs/downrev.xmlDQK0Co.uc묻c* V][mĮp[z݈A% VզDx | Z-(SdB)"LJ_Tɏl&vG4])SFd&5&>Ttw[O&]_߂܇M/I!lz8|Zv/FSPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Pdrs/downrev.xmlPKz9 <$0 g The inequality |X:X Xg|d" |X,Xg: Xg | proves that any strong inert subgroup of a group is likewise inert Thus strongly inertial groups are inertial It is easy to prove that any FC-group is strongly inertial   8E!E!!!!&!!!!&!!!&! !9!I!!!!A  p A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3fF>___PPT10..+p3g+JD' = @B Di' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*pQ%(D' =-g6B fade*<3<*pQD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pQD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*pQD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*pQ%(D' =-g6B fade*<3<*pQD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pQD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*pQD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-g6B fade*<3<*pD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*pD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-g6B fade*<3<*pD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*p+8+0+p0 +"'T2 pt(  t t  x`a0e0e?Rectangle 3" PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!@@qdrs/shapexml.xml][o"7~`BBVI+TX)3m_Შ&PFs؟c\~^{ڤOuΤJqzǻ Όj,B˲kJtZ[vk5Le.̧ &΅ţ~Z(jz]ERr]r<,ypDld!2x /E/(wr0UQSYNֺxJ165y*d)̾rT_9_xivN#uY/.&:W[L&g}n8{YLt²/Nh6V Rʮ]ܠL.qW6 JFS@oh6̅e_s 8:oAo~m~Q\#!cѡED͢IX11Y\Y`4f 4,&hX|b(ZD1>V4+sHT4+fe41_]hcE74+ j&gQNhרxpDlRB x ר>D.w+C\غjƢ7$}ỷ3עB_(c&Vx~]q:"حAH@.w>x1mѩE@~ǰsoFX!~jpJ\:}Ӱc-сE<#\cs t3cqr̛hFavڻ|DPZB>wUFyu _ݱt,%KC%Xc.lQO?DžL4d}5?dWwCHbYGFQATہ}kl F=bwVӎD8G`N:b1E(-ԋ ƿ"^q8[O"!rk&_C[r WIIlX>&ͲAtl,Zarho&,CL8d$faS,"07ԔQk0ǿ]׆a&qX|$Mw766Ctw:; $le BCgz'#[x2h +jWV*xΙ\oKW9CAIԎ*BN+va`(+ kA5mKR+Su\xN+߆[-!e5kh-{7ZUm[#?t2H&LֵaC[f߯ba!(<XwD6XeVA,˼hd WGm6<)ܶ:eւBP[p˂_Lchw$ԏVC欒5uzrqПą|Ym0ӑط I3HCPedҸ"uĊ}C&y|G,d}KC@N=$-; EADCƘ[5^4).~ݲ̪ ;h&ZtiOH})IP(4s=W.25o?Obltzb=TAC*OڎG:g9ݛO6i@|˜cP:8w|_Q8|"2; ` PK!e@drs/downrev.xmlDN@M|͘xcdLL>bw]Ky{7^9N٢7ڲ A\X]s#Yccb~y1LۇRD T!Ƞؖ8v ])SF%D9>TҪkmiOUnGM~"Pݼ_Zki2߃8?]w9/FSPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!@@q*drs/shapexml.xmlPK-!e@ drs/downrev.xmlPK  N<$0 g -Clearly, any normal subgroup of an arbitrary group is strong inert and so inert On the other hand, finite subgroups are inert but in general they are not strongly inert In fact the infinite dihedral group is inertial but it is not strongly inertial Note also that Tarski groups are inertial r-  G! 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Robinson, 2006) Let G be a finitely generated soluble-by-finite group. 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E+JD' = @B Di' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|-%(D' =-g6B fade*<3<*|-D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*|-D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*|-D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|-s%(D' =-g6B fade*<3<*|-sD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*|-sD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*|-sD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|t%(D' =-g6B fade*<3<*|tD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*|tD' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*|tD"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-g6B fade*<3<*|D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*|D' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*|+8+0+|0 +"]2 ~(  x   xG0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ڽidrs/downrev.xmlDN@M|͘x'[jBbHH>jw]/|'d6NֲQZGzYcg\lz}5B3*hB )}ՐA?=q 1Zj7|Hgi`OQge墴3v046+QZA)ZC)~6PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ڽidrs/downrev.xmlPK'gx<$0 g P Neumann s theorem cannot be extended to strongly inertial groups. In fact, the locally dihedral 2-group is strongly inertial but it has infinite commutator subgroup $ !   A$??$ArabaRossapiccolaPicture 4ArabaRossapiccola0 +T3f3f  ___PPT10 ../+JDZ ' = @B D ' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =-g6B fade*<3<*ED' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*ED' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*ED"' =%(D' =%(Dr' =AB,BB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =-g6B fade*<3<*ED' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*ED' =+4 8?dCB#ppt_y+.05BCB#ppt_yB*Y3>B ppt_y<*E+8+0+0 +"3 (     x@H0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Zǜdrs/downrev.xmlDN@M|͘x'[jIe jLH<bw]=/|'d6NQMeUkj]v7E5pfՄreOfmEBBK髆5nortpɇ$yZyp}z]t\.Jן7D!RE}()KSkՆ|`m49PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Zǜdrs/downrev.xmlPK'gx<$0 g r Theorem 14 (M. De Falco  F. de Giovanni  C. Musella  N. Trabelsi, 2010) Let G be a finitely generated strongly inertial group. Then the factor group G/Z(G) is finiteb ! !B!k!   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