Question

Let N be a normal subgroup of G and M be a normal subgroup of N. If N is cyclic, prove that M is

a normal subgroup of G. Show by an example that the conclusion fails to hold if is not cyclic.

If is a homomorphism from a finite group to a finite group ′, prove that |()| divides the

gcd of || and |′|.​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

Let N be a normal subgroup of G and M be a normal subgroup of N. If N is cyclic, prove that M is

Let N be a normal subgroup of G and M be a normal subgroup of N. If N is cyclic, prove that M is a normal subgroup of G. Show by an example that the conclusion fails to hold if is not cyclic.

Let N be a normal subgroup of G and M be a normal subgroup of N. If N is cyclic, prove that M is a normal subgroup of G. Show by an example that the conclusion fails to hold if is not cyclic. If is a homomorphism from a finite group to a finite group ′, prove that |()| divides the