Question

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

Answer #1

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 H and K be subgroups of G. Prove that H ∪ K is a subgroup of
G iff H ⊆ K or K ⊆ H.

(Abstract algebra) Let G be a group and let H and K be subgroups
of G so that H is not contained in K and K is not contained in H.
Prove that H ∪ K is not a subgroup of G.

Let H and K be subgroups of a group G so that HK is also a
subgroup. Show that HK = KH.

Let f : G → H be a group isomorphism, and K ⊂ G be a subgroup.
Show that f(K) ⊂ H is a subgroup.

(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.

If H and K are arbitrary subgroups of G. Prove that HK
is a subgroup of G if and only if HK=KH.

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
.

Let G be a finite group and let H, K be normal subgroups of G.
If [G : H] = p and [G : K] = q where p and q are distinct primes,
prove that pq divides [G : H ∩ K].

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.

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