Question

Show that A-n is a normal subgroup of S-n. Hint: what is the index of A-n in S-n?

Answer #1

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.

Show that S n is isomorphic to a subgroup of A n + 2.

Let N be a normal subgroup of G. Show that the order 2 element
in N is in the center of G if N and Z_4 are isomorphic.

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?

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.

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.

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

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

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?

Show that the commutator subgroup of GL (n,F) is SL(n ,F).

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