Question

Prove that A_4 (a group of order 4!//2 = 24/2 = 12) has no subgroup H of order 6 by showing:

1) all squares of elements A_4 must be H since (A_4:H) =2

2) All eight 3 -cycles are squares and hence must be in H.

Answer #1

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...

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

Prove that a group G of order 210 has a subgroup of order 14
using Sylow's theorem, and please be detailed in your proof. I have
tried this multiple times but I keep getting stuck.

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!

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

Let G be a group with subgroups H and K.
(a) Prove that H ∩ K must be a subgroup of G.
(b) Give an example to show that H ∪ K is not necessarily a
subgroup of G.
Note: Your answer to part (a) should be a general proof that the
set H ∩ K is closed under the operation of G, includes the identity
element of G, and contains the inverse in G of each of its
elements,...

Let G be a group of order 4. Prove that either G is cyclic or it
is isomorphic to the Klein 4-group V4 =
{1,(12)(34),(13)(24),(14)(23)}.

3. a) Suppose that H is a proper subgroup of Z that contains 12,
30, and 54. What are the possibilities for what H could be?
(HINT: You may use without proof that all subgroups of Z are of
the formnZ. We will prove this fact later in the semester.)
b) Now, suppose that H is a proper subgroup of Z that contains a
and b. What are the possibilities for what H could be?

Problem 8. Suppose that H has index 2 in G. Prove that H is
normal in G. (Hint: Usually to prove that a subgroup is normal, the
conjugation criterion (Theorem 17.4) is easier to use than the
definition, but this problem is a rare exception. Since H has index
2 in G, there are only two left cosets, one of which is H itself –
use this to describe the other coset. Then do the same for right 1
cosets....

Let p,q be prime numbers, not necessarily distinct. If a group G
has order pq, prove that any proper subgroup (meaning a subgroup
not equal to G itself) must be cyclic. Hint: what are the possible
sizes of the subgroups?

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