f Slides 3.3 Normal subgroups is the set of elements that normalize - PDF document

Sec 3.6 Normalizers and another application of corsets not normal in G Idea If H L G but H G measure how far It is from being normal we want to H OG iff Recall the Def V xH Hx X E G how far It is from being normal One way to measure X E G e It satisfy is to check how may H H Think of each ett as voting yes or no to the normality of H x E G vote if xH H YES NO if xH THX Remy Every c H votes YES Why EH XH H Sec 3 3 Every X E G H G G votes YES iff not normal there is at least one elf voting no It is If in favor of His normality Defy for the EHS X E G which vote Yes Thehormalizero fHing denoted NG H is set x c G x H Hx the xHx x c G H f Slides 3.3 Normal subgroups is the set of elements that normalize He wording we say Ngat E Ng H Propt xH E NG H then If Prop 2 from slides 3.2 Cosets'D all ye xH yH for H so it doesn't matterwhich coset representative or y X you choose Hx Hy for all ye Hx for the same reason xH Hx by def of Nc H Then x C NOCH ftp.go Suppose Think to self Let y E X H my goal is to show YE NG H ie I want to show yH Hy Then yH by above lemma H Hx by C Hy by above lemma D that members of a Ree Prop 1 tells left coset vote together us vote yes G hen xH Hx members of a block at x H as hen XH f HD vote no at or

PropI Neat a multiple of IHI is PI us that NGCH consists of entire left Cosets of Prop 1 tells 11 at least one H itself we know the left cosets are the From slides 3.2 Cosets and disjoint size same Partitions of 6 Cartoon example by the left cosets by the right cosets H Hy H y H ZH WH Hz xH Hx Hw The elts of the cosets H and xH Hx all vote Yes The etfs of the left coset yH all vote No since yH f Hy 2 H n wit n The normalizer of H in G No A H U xH is The two extreme cases for them Nc.CH are NG G CH iff H G G cartoon ex when It is as far from normal as possible NG H H H Sf LG D6 f r Ee H Obf The coset r3H is the only r It itself i If 7 vrolerfeatoheodsetfr.fm fyhichmorcaennoIhanbeone it of DG fK ett r E g aanndbebyreaffed from H by r r ra yr i L T Carr be reached from It by r4t r and r4 L C NG r3H r You can check CH Hf H Hr3f rsf r3ff r3f f Pf and the other 4 left cosets of It are not subsets of Nc H not normal in DG fyi f is

f So ND LH r Lf U f f e r f fr r f e f r r3f r E th f Exercise a pattern for Npn on whether 1 Find it ndepends fy odd is even Thm subgroup of H L G G Then NGCH et PI we need to check all 3 properties of a subgroup being x EG I Recall Ng CHI Hx H he H Contains eh e e HE H Inverses exist XE NG It Let We need to show I'c Neck 1 E H that is show HG y i Then 14 1 with x THE 15 Hx so replace x HH by x e He H Closed under the binary Suppose x y E NG CH G operation of meaning H and y Hy H x We need to show H xy HEY xy E NG CH meaning But xyHCxy5 xyH5 x shines 5 4 Ig x H x by by 1 Hrs my core is normal in its normalizer in G Every subgroup of G H L G H 4 NG L G If CH then a normal subgroup of NGCH H is If