Question

(a) Show that H =<(1234)> is a normal subgroup of G=S4

(b) Is the quotient group G/H abelian? Justify?

Answer #1

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

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 that H is a normal subgroup of G and K is any subgroup
of G. Define HK = {hk : h ? H, k ? K}.
(a) Show that HK is a subgroup of G.
(b) Does the conclusion of (a) continue to hold if H is not
normal in G? Justify

Let G be an Abelian group and H a subgroup of G. Prove that G/H
is Abelian.

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 an Abelian group and let H be a subgroup of G Define K
= { g∈ G | g3 ∈ H }. Prove that K is a subgroup of G
.

In each part below, a group G and a subgroup H are given.
Determine whether H is normal in G. If it is, list the elements of
the quotient group G/H. (a) G = Z-15 × Z-20 and H = <(10,
17)> (b) G = S-6 and H = A-6 (c) G = S-5 and H = A-4

(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.

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?

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