From a2d9ee6323febdd975ea5442deb81c4b7fe3a898 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 18 Jul 2023 21:06:26 +0200 Subject: [PATCH] fix(algebra1): migliora l'enunciato della prop. sui centri di un p-gruppo --- .../3. Azione di coniugio e p-gruppi/main.pdf | Bin 157530 -> 157574 bytes .../3. Azione di coniugio e p-gruppi/main.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.pdf b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.pdf index 83369a3f9b89983fdcd1c2de152f4cf487c72f2e..57dcf80788973faf2011e311e2ba4389b37efa10 100644 GIT binary patch delta 5407 zcmah}c|4SB*k)wkCE3QBC8777eH=THEKw*Sd!>-cz8o*vcV(MwSt>%-$d<@fDWsEq z$(lo!$;ev1>3rv$?==4Sp7~?ucg=l2_jTXb_1q&p%n`lJ6&KiGXk4MdyC7PmA3Zg7eADTTL3y)=K={HjlkX0i+nnG`#6iS(xYlu_J`6lzb=n9T^bzJ z*RZaA-L_vGWI13KBI;7Bs5Ju>7nRk&{${vhGcssMVh(Kct&LS#;XHZ#^I(5NDaUHr z;+oL}Z(g^)^sH;M9Cj3Ui4@saJ=XrXa-%{aqway*H^b&ozo3Tg?%RrKY~GSyL-gO0 z(x$EErpii5hnFsKnr%UY!Yixg>0?*W^_MSQtx~)2Y&*#2r3kM_5a&=iG_f2rKX^l` zTA5>1*xYHN99ugbIY??7oO@w&?i7J=KS*J2ZHPYSnOVh^7XyubR|C-*#^R$Q02gb) zbds-JUFJ8_GK&GSPsT2JvEI~!Zd zbl7E@#e3KO*^LY5m(SqRXW#z8c5^DSuPgW^*Hzgz#Wds2DPlC?>np!j9Eae}WsiJu zW>u?W{9^Sni{`i5zU!Z1uJJEDtPza$<+FPBa_1ftLePr)H1k9E7!e8O;~U9m>MDn; z`z|cKcTaI29A9(2u0yXovEiF77}lDWdZTe|HXK2WG&~l7V2Ko382IjSPAw`e^m%Ai z4Qqxhvv9~#*`mcQ4PkOo=JLk)+n6|R-QJPb+A;dW*-4Isv}3U!?#e5M)tzS{SzPC@ zz{H%Ev|CP-?baMTiOFn$Uek8f9~Q80dL6@%ciDfhb)P13R9QLSu4TWMlf9+2A!>EG zzrvQD@ZIjm-3cN7rq&4~(f3&7(G%_3(J9V^c*!z$A56k1hf(NwqdV-bJ1f_baP zK~F!+Q8!qZt2MFU=!$_+i~iXJW7-)94vyn)tM{3KTy0rn*|Ml>Cyx~a3l}C0d0NSb z2)Z+}!DoK=?5Mk1LjzT7-Bb2i+E>_`UWM7rEXRkDK0mcPY#_{ta}$uy46(p!cX&au z2BxpR)VkFUs@rmA4pk^(ioDK68GZiUV=r+w(@8-lBI3~)9J9N1%OV`Rx_-r`Y%#uY zVaT1PJeTQrIbv=$YvhRa$&haGJEPmWvz`-YnM(BXV|Z<~fAfHX?vHqtV4J%T{el9e z>rsz=6J*Aw~j<~SrdF@H`U3ZYjI`F&lx|>Di^n>6H-2{Jy@CG z*6F~XZa?c*^&!z|DS=qJGUwc6%A16Nw-yZT}S7{mA!J&W%SNl1-x%k#fZuB?Ke z{Q8i0gK_stfr&3T(eAxE%#DD!+rmK$Ca0x2h9hump+2*fmF%#O z<5tm0Z}ZONC_&dN8(VndxR_2h6q6nuhX?Ggedqf|)Lny8FJym?b!X31SYKco9Ob7Q zS)NH+@@-zLD``t6dAp8xp8aNRd_uSA#{(g`p33mn3|zbPzC68=e$VsH__DpmM+sat ztOGLaB-q4kA$Ym+yos}sfcdl6)xdwZ^uQE{7GV}N0_M62<0~N6bq|)>5 zeAzRrbPbK`4f{@=ax7066wQ*#X4wfXhcO#Eg%wW20l^MDGt0&|UD0GOF^$EWoabXN$e8{B?^s0%YO;y~!Naop?=vZE`g9WPky`L)q1A zT93RZZB~$rVq_x7!Gd|^?LqCq}qqw z4Eyw@@~xSawzu>q!8LsnDfc#s>3yy1vTOI?M?UYoh01vxz`jHg59su@0*2eExc2w5 z>Sp7p7&QUy4UNHiN$ovG=0lbb!+m>UxP-?vA(D_0UFS7gwq#M(;^57NcbzT<4~6sz zqL$kbo4h(_1VP?t`2E81@Dh4H3H2V!z`&{F;@Rgk?~jG-y9kT6oMdPMqL=MaRPyMIoiwgGmdGoxPpkL?x+#WJ>h>&K36;K$-= zKWa&xQ`hHks*~?k&9$>G<;=|}kq8pqz&ev@p}_D4;zOuNR>}I#qRJC5(uth&+$ON1 z5O4pxj=Hbfhf>drUz;~oFKuw8cRUfOs2rG@BGe+f_=n#VJm6SULCf{T0!b#Dl)ezg;gWX+xeow)*K#^BFj=XaTyK!;G3B$2L z`Owv90`LqoRTE@=c99V}YhSKqNd8ujI$BX*J21E>eB?lZ7)DlQuds8n$7V99N^~ zn=-#wq)86;aWxffxp72N0aOp{Pql|-K}Q++54q3JGA_1mDNBtp>o>Fk`73-n<>_(f zlp56R;ke&XKtE5p0rC%xW({7r;hQWX<^r6qGhIEE`E^fNK_x*F^w+f&x#7e_HS zt8iCUpKP@)wjG{aMhhJ!HYK%fZw3BjS|S18mJ?=|4jD$M z+iBCVoWy>-?r~8!R3R=T`LM%-MU|tXuc8?ypzExUGPdgF%xxzsi(UDTY;sSmi3$LI z3&V13rY4m%3$Pc~MkOrxv4@kk#Mrdy1DBY; zSW5cM*=?8Q<4N2$vh%Cc0ouWd>JUkPl~XBW97{BGn^||V3}In#_OY2a`!WYN+T!Gp zPHW~hgHNt32CXIcGPUlD*S4rTHN9b2t$ksh^)mU&(IxE}52bn&>{69ne@D)Ql2K6> zUT4l1p(4OSmNs=u@&pPVXw~bhr25SujgTvom8 z9+%Tvy$lnawn7v>s<^VWnR;H!>*H~J+BtjKDz>w)<%>|--z@c5P|8)!`FT0fb)Z6! zBv-l)M+$?_^ohNx{dYBU+(fO=^69&ByO{lZG$I%@ZmUw<)a z5udN>W%@Jfyf}Qd8N3EjytiZA!DG@ILN;ifdY4CJ4k;30 zp66c_Q$bn8*M!K7ygIBFx$dOc?bE1L-Ff7UTxFW9XZ)8Q$5v7WUGIlZIr&(dnI@%QGOKM0UOFGFV4${Crv+_Z*S zs*p1%G$+D)-&3Xe%!&9#=(P6B+s(u$S{)gVo7&uM8sgFTX{>YUuw&w8W>RyMo?oJk zrl&fXon(%7vh_JDl^Hsjty1I!-&^@YcGtE4&DOW2R$2OKTNbTPyqfgW&{*Ss73rrQ zf2z(){$@b8;q47^4U?D|FmW&3yPj{^ujlO)jk(2o-CK3g4}v(2Mfa}lS4+e?4gYv( zCaUHZ*>Q<({}aSc|GUl?ffsd8O3)E2X^)wDRB!6ntzlW$YcIxSU8B;qIW}tGxKyyzxeRbgFbnM?EFAFFu2HlGW2pDES&iaKie}?4Y8e2 zP4YG;a-5_diWiK;3(`Kr3&N(f84w_VLSxd!Y+&$zerwslb|5?gBL@6DVs2tAKfgf~ z62zjZF$^Am00Yqg?m!#>VyQ4R2w)D>Km#b?XB_n3!UzNkq)tX402CF5!r@WW7oyPs zK!s5NJPQA7E(OE>(p>s)b}%6b{}-44P(h;*804?1AQ}ncsp+9m;DH&SQMd#2P*^;c zDi;8-AO`VQ4d~zD{{(Rd`oJQni%{Ya$p20LZ!U(?!cPr zkVq=W07wLgrm`gkLj!-i1Nz7Nz)#t!8KMw4JhhF`0HqhIQKJFmfeE5<7&P@{&^R3K z*Idf{u_*lCX81qtuxKQXx)X|WNa~JpC@hv5!-H5n^|bH^3TrCTcsvG;gARarkZSxC zn_~~W2mlX&R7F4#cktyfhYilrIHw;fEX%GD0>jZV1Ct!f+3IxFGNxfbrFh@sB{NWXb?{&6Ntk6+_itb zzkhWHQCN`L{U8blA`Xzjg8;Qmlr>2GJV7)_S*%n9(Fi2wUoSv^Z+eN{smj3~OOk*Gb delta 5364 zcmai2XH*kw*QE(Ukd72l=|w4-WRgroL7IgkQZ8UYdX?U#4MhO~K`GLtBcLb{P!I&9 z2o^w8q$6E=Xo@Icc;oxM_pa;kDfmZ$bIumE3#HbDb$s}Yp{Qk@*}|` zSnP58{K`@x+wznSXKkIe>d174M|FHY?+et4cGG);m*ThbZfecjjee2q_&OP*{CL)} z)106mX=={yAoK38L(6Xkl}QcjCO39`J69<7FB?mjS%?P ze3e{!wrf(y8sy|TobqSVt=Pz+GGmqaOd&TwYR#Mp`jilXSFpPyRN~tsJU`^NS2)(C<6{?1v;Kt<)=uXj82UtF%6e!?aWW^5`<08-3)$|oXv<2zo35$l zc4tp#x9HJkhUd@1zMp^MT%G<^?W@XjG^n4MJnt%`KI)=?svh5A`T9~^K z$Xb8bP>n0_nI>@3leZ^xI^t_e5|`upw{r}8Msj7JEhVjEcjzV~_?wD;>Dr_1)L~B_ z@*Ov;%A(w|-o$a~@lxgSp--_a>LVL|nS$4w?j|qPu1tkuQX@OT06a@1_iW!UgSB7| zxzhoW?Y(mLK4(udlqx?gdb7v<+384#@b$|5?eLWW#3(`c*ZsH z&NQk^za`|Up1L6O)oS$%`)h+qLp?e$8}u&oHOHim5x95#<0LzU2^yXm=iEn6fqk54 z*s3%F%XC`8cwg>o-nR31HD5Uups0}Nx}VJ`YCuB2umuD&#arlWsoWlJ>hstuFQAkX3+5HF{_(fR0s~lSQ7#Slx?>s|-C{ zmYX3SHYYG~u1$|7d2=e-y~}YAY#WKWHrKmzX{B{^Peb8QhV-PaLyKf8w9joAV}hrz zbcu>ypzKd8zCGaldC@hb@sKac;?m+u?^%&e#pR4iSbS7=0RlkKCSdsBM7s^|NL^Qk zTFnFBtURk0&blzsZx77FT;zow)?c7cQ4{UkJU!(#qPw@`Y+f{vxq5&n9GpGq?duvC z?_Geyn9MNitco8JP^?IIUghF>aO4D;p=0$ZYrM0&uPpn?Z{cg&am@8hV^hbk+*Fv9 zZW=$ESuYkFt#vyp`?EEy+?%OrS>gY_XiilkRwe+Hd;rLCRxrJpvB^^RJ}vEAB%>td zFRVHXxR+FZ;8IiL@s;YGgTXkih0-)h6KSZ;!{B<+@X)IGzV4TLC%BglFB%+-zRH)A z^&t7o=_j)qLunKNK6)$2?ZpPo98t0(SA%`22a_F8{!`}L#0YKf4OHes@v1Y-b_}Rf zYu1kz+hPJ`G=4QT3W9ZG}+4;9%e=LC#Lg-4btAl#}R{mwN3LGvPXn7SX;3z(Iy-43x;v33O@7y zImAVf`VjL~Pe4YLk`c1g&#`*=u;TVcr;T6bOrY%7$-zep0JF!BTq%(>$eM4Wk$vRO zt2X()h8vy|s+dBMb6 zFO1{pVO&Uz@S}BI`^sO|Un9-N?a#2PHDfhO4&3WO!)r#+f;37fv4(KdN9-cjxH430nHYnX?hJ$7u=$lf)uyU&z*9{4AaRMJkDYM96%x zT_w5b{P-LF5ElD|T)K?*a~X;o@4JioH;=*m%IPs@1>n-z_ZP^AnoN|Ua$;k@OsED6 z$UW~}uY2#1YDOdo!K<%il;cr(G@>~g(v^#|k8{HbJ7e?2Bl6zsl#3E=mW*denTifp z8Kt3o;@2U8J^Xhwn6mA{i$#o{iK||hdw(*-MDC@+qmb_NG%IWqPcDa!(^|gNZKZ3R zTeYv|XPJGaKOD(G;rnGV+l1b3mQfV=mhbk&{P>>PYmxC?ClcXYPsvG7u9a+^&lk5X z)$J^=(D^^ra7zfHV^umFZEI#k>aL3Vgfb9R3XU>eTWYsoPfL9{yY2g>%jQzU5qhGP z6tRjYf>qOZ#8W&oAeK~n;S;Pb`ZTt(*QOt$gnCE6PUkub6j$V>9bVj+0I%{U1 z=Wn|^{k$lhxhEZtv@qW`Jas`RdVfLUprYTYj(g9oCn~fG;hxl=#|tH$N6!{T5t=;I(R|FrG5K?U79>VpwNp;J$}bdipx?miyQRPt)vM zdBYCIk(_UVStadnAP^_I7~1}%E}{6-UBY>7~k>K_y$Nlt3} zP3CkZ%IPqkg&%&k)}1x-*i-7E7pmNF>qyO3@wbthO#hks21bd}<#4HJ74MZtPcj7C zgUHU!-=AqkM2R2Q^sK=Owa_{9#WJo8zNAr!y_)5HKb2;tm6nnA9`R?WKU%4t(XKgx zf{rcTy73_cm!Bjr&%s~+J|(*`pdh(sx-qCQVMmR8Viq>G;a$iNA=DH1i?}g#0cjtD z7#<4P^XZqDIuBU8zYD}KhTn?xvqa52O7Kl>4qAwh+vD)VzcAP1i;72VoYnpK22)g< zHG}PC`En_Xe(yw)^ZcA4%?mnD2M$($Fh^OP^stOJ44;{xCw^$$(lyQ0H;W4to6SuKt23Ppl^&37SJh@|! zGx?kSPMu7N4Eam?$cV$RL_dsk-H?S z@7!@SbL%i^v0#J2F|l5Iyd>j~{gWH~>dPQq%iJ87Fr?*D_Vv}klS2EuA?}TVrEZN4 z*P<`OiZTvt2P10-PV;#m(K^Q5tiBYWlB#tNSN zzmxlo1Wj2K zi>cUi>;<>ui?0tslg)$t?Aw`#m(-n@ zSlRpUZZMP4c8#W!%iM!a8q#wkQcpa|N_WK+wdvP=e0T;{&+_5pUbwxg-?aQn+AJPk zj&V*qY^fCtv!6e_ZNGyOiQ(;~-0$QtScW57H&Huda6dzk zVXXtgUQxd8*C(f)M-(JZ`REu`tv9&1ETn$$2;)Dt^}#jkh`!Iu;zVV}mr}AUVo7fe zWIp!miRy8c0>In#xxM?89*qY(_WZOFK_@=S_MtFo&h+6o0cXr(lwiB9a?O1$-l3&B zP@h}rq#?AdS9U(Rg+YQf;!;TtM|ciL5QVL!Y;xrzrO&81+V}T!xM44FaX4Z$T%F4fJI<90FBvo2S_C1 zf5*Z9z8C`#3CLs&fJY-R5F(tONole5VJe76XhJm>fXDo)6BWZ?cOQtwK*)`V?2%(`}WT@*68AmlHay$VXRTPAe02~Gj z{Nn}qf14kN#UkgEN*PP|C-Cn$s?~R60CpD{9My%0LU1??;a}bI^007lO4+cbL!04L zsV{_6)u8^5K?Ky)0UUwaZ2+SV;nj#36(!;0*#9rYR9H!g#na2e!^_XZ#+C)ZQ6CJ8 LgoK9fDVF~LIKfZT diff --git a/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex index be612c2..cd46e95 100644 --- a/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex +++ b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex @@ -54,7 +54,7 @@ Infatti, grazie alla formula delle classi di coniugio, si osserva facilmente che il centro di un $p$-gruppo non è mai banale (ossia composto dalla sola identità), come mostra la: \begin{proposition} - Sia $G$ un $p$-gruppo. Allora $\abs{Z(G)} > 1$. + Sia $G$ un $p$-gruppo. Allora $\abs{Z(G)} = pk$, con $k \in \NN^+$, $k \geq 1$. \end{proposition} \begin{proof} @@ -67,7 +67,7 @@ $p$, si deduce allora che: \[ \abs{Z(G)} \equiv 0 \pod p. \] Combinando questo risultato col fatto che $\abs{Z(G)} \geq 1$ (infatti $Z(G) \leq G$), - si conclude che deve valere necessariamente che $\abs{Z(G)} > 1$. + si conclude che deve valere necessariamente la tesi. \end{proof} \medskip