Question

Let N be a normal subgroup of G. Prove or disprove the following assertion:

N and G/N have composition series ----> G has a composition series.

Answer #1

(a) Prove or disprove: Let H and K be two normal subgroups of a
group G. Then the subgroup H ∩ K is normal in G. (b) Prove or
disprove: D4 is normal in S4.

If N is a normal subgroup of G and H is any subgroup of G, prove
that NH is a subgroup of G.

Prove or disprove: The relation "is-a-normal-subgroup-of" is a
transitive relation.

Let φ : G → G′ be an onto homomorphism and let N be a normal
subgroup of G. Prove that φ(N) is a normal subgroup of G′.

Let H be a subgroup of G, and N be the normalizer of H in G and
C be the centralizer of H in G. Prove that C is normal in N and the
group N/C is isomorphic to a subgroup of Aut(H).

Suppose N is a normal subgroup of G such that |G/N|= p is a
prime. Let K be any subgroup of G. Show that either (a) K is a
subgroup of N or (b) both G=KN and |K/(K intersect N)| = p.

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Suppose : phi :G -H is a group isomorphism . If N is a normal
subgroup of G then phi(N) is a normal subgroup of H. Prove it is a
subgroup and prove it is normal?

Let H be a normal subgroup of G. Assume the quotient group G/H
is abelian. Prove that, for any two elements x, y ∈ G, we have x^
(-1) y ^(-1)xy ∈ H

1) Let G be a group and N be a normal subgroup. Show that if G
is cyclic, then G/N is cyclic. Is the converse true?
2) What are the zero divisors of Z6?

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