Question

2. Prove that if f is a homomorphism of a group G into a group with kernel K, then K is a normal subgroup of G.

Answer 1

SOLUTION

TO PROVE

If f is a homomorphism of a group G into a group with kernel K, then K is a normal subgroup of G

EVALUATION

Here it is given that f is a homomorphism of a group G into a group with kernel K

Since e ∈ K

So K ≠ Φ

Let a , b ∈ K

Now

$\sf{f(a {b}^{ - 1}) }$

$\sf{ = f(a )f({b}^{ - 1}) }$

$\sf{ = f(a )f({b)}^{ - 1} }$

$\sf{ = e_1 {e_1 }^{ - 1} }$

$\sf{ = e_1 e_1 }$

$\sf{ = e_1 }$

$\sf{ \therefore \: \: a {b}^{ - 1} \in K}$

So K is a subgroup of G

Let a ∈ G and h ∈ K

Then

$\sf{f(a h{a}^{ - 1}) }$

$\sf{ = f(a )f(h)f({a}^{ - 1}) }$

$\sf{ = f(a )f(h)f({a)}^{ - 1} }$

$\sf{ = f(a)e_1 f({a)}^{ - 1} }$

$\sf{ = f(a) f({a)}^{ - 1} }$

$\sf{ = e_1 }$

$\sf{ \therefore \: \: a h{a}^{ - 1} \in K}$

So K is a normal subgroup of G

Hence proved

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