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Posted: **Thu May 12, 2022 7:54 am**

: Notation 2 3 Let G Be An Abelian Group Then For Any Integer M We Have A Homomorphism Pm G G P G G Notice That In 1 (38.59 KiB) Viewed 21 times

Notation 2.3. Let G be an abelian group then for any integer m we have a homomorphism Pm: G →G,p(g) = g". Notice that in order for this map to be a homomorphism we must have a"b" = (ab)” which is why the condition that G be abelian is necessary. Notation 2.4. We set Im = im(Pm) and Km = ker(Pm).

= Problem 2.6. Again G is abelian, #(G) = mn, (m, n) = 1 with Im + wn = 1. We have Påm: G + G. By defininition im(Pim) < Im, so set Ym: G + Im, Ym(a) Pim(a) and Yn: G + In, Yn(a) = Pwn. Set 7: G + Im x In, 7(9) = (Ym (9), 7(9) Show that 007 = IdG.