Question

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.

Answer #1

**Solution:**

**Given that:**

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...

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 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,...

f H and K are subgroups of a group G, let (H,K) be the subgroup
of G generated by the elements {hkh−1k−1∣h∈H, k∈K}.
Show that :
H◃G if and only if (H,G)<H

Suppose that H is a proper subgroup of G of index n, and that G
is a simple group, that is, G has no normal subgroups except G
itself and {1}. Show thatG can be embedded in Sn.

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?

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.

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
G be a finite group and H a subgroup of G. Let a be an element of G
and aH = {ah : h is an element of H} be a left coset of H. If B is
an element of G as well show that aH and bH contain the same number
of elements in G.

(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.

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