Question

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

Answer #1

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?

If H is a normal subgroup of G and |H| = 2, prove that H is
contained in the center 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).

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

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.

Show that if G is a group, H a subgroup of G with |H| = n, and H
is the only subgroup of G of order n, then H is a normal subgroup
of G.
Hint: Show that aHa-1 is a subgroup of G
and is isomorphic to H for every a ∈ G.

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.

Suppose H is a normal subgroup of G where both H and G/H are
solvable groups. Prove that G is then a solvable group as well.

A subgroup H of a group G is called a normal subgroup if gH=Hg
for all g ∈ G. Every Group contains at least two normal subgroups:
the subgroup consisting of the identity element only {e}; and the
entire group G. If G=S(n) show that A(n) (the subgroup of even
permuations) is also a normal subgroup of G.

Prove that if A is a subgroup of G and B is a subgroup of H,
then the direct product A × B is a subgroup of G × H.
Show all steps. Note that AXB is nonempty since the identity e
is a part of A X B. Remains only to show that A X B is closed under
multiplication and inverses.

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