Question

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?

Answer #1

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!

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.

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 G be a finite group, and suppose that H is normal subgroup
of G.
Show that, for every g ∈ G, the order of gH in G/H must divide
the order of g in G.
What is the order of the coset [4]42 +
〈[6]42〉 in Z42/〈[6]42〉?
Find an example to show that the order of gH in G/H does not
always determine the order of g in G. That is, find an example of a
group G, and...

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.

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.

Let G be the group Z3 + Z4 and let H = h(1, 2)i be the cyclic
subgroup generated by (1, 2).
(a) Find the index [G : H] of H in G.
(b) Is H a normal subgroup of G? Justify your answer.

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 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 G be a ﬁnite Abelian group and let n be a positive divisor
of|G|. Show that G has a subgroup of order n.

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