Question
Asked Sep 28, 2019
4*. Let f G
H be a group homomorphism. Prove:
(a) If S
G then f(S) 4 f(G)
(b) Show by example that S aG need not imply f(S)
(c) If T H then f1(T) G.
H
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4*. Let f G H be a group homomorphism. Prove: (a) If S G then f(S) 4 f(G) (b) Show by example that S aG need not imply f(S) (c) If T H then f1(T) G. H

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Expert Answer

Step 1

To prove the stated properties of group homomorphisms

Step 2

proof of (a): the image f(S) is normal in the image f(G) . (note that f(G) need not be H)

f:GH.group homomorphism
so, f(y) f(x)(y) Vx, y G. ()
(a)recall.a subgroup S< G (normal)
if gsg SVg G,s E S.
From () it follows that
f(gsg) f(g)f(s)f (g)" e f($)
if gsgS
Thus,S Gf(S) 4 f(G)
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f:GH.group homomorphism so, f(y) f(x)(y) Vx, y G. () (a)recall.a subgroup S< G (normal) if gsg SVg G,s E S. From () it follows that f(gsg) f(g)f(s)f (g)" e f($) if gsgS Thus,S Gf(S) 4 f(G)

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Step 3

(b) example for S, G , H and f, where S is norm...

(b)Let G subgroup of H S
the symmetric group,with G {e,(12)}
Consider f identity map(of subgroup G)
f GH S,
Let S G.clearly,S 4 G
but f(S) G not normal in H S
for example
(123)(12)123)1 (123)(12)(l32)
f(S) G.
= (23)
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(b)Let G subgroup of H S the symmetric group,with G {e,(12)} Consider f identity map(of subgroup G) f GH S, Let S G.clearly,S 4 G but f(S) G not normal in H S for example (123)(12)123)1 (123)(12)(l32) f(S) G. = (23)

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