Question

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].

Answer #1

(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 G is a group of order pq such p and q are primes, p<q
and therefore |H|=p and |K|= q, where H and K are proper subgroups
are G. It was determined that H and K are abelian and G=HK. Show
that H and K are normal subgroups of G without using
Sylow's Theorem.

(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 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,...

Suppose that G is a group with subgroups K ≤
H ≤ G. Suppose that K is normal in
G. Let G act on G/H, the set of
left cosets of H, by left multiplication. Prove that if k
∈ K, then left multiplication of G/H by
k is the identity permutation on G/H.

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

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

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Assume that H and K are subgroups of G such that K < H < G
and ( H : K ) and ( G : H ) are both finite,
Then ( G : K ) is finite and ( G : K ) = ( G : H )( H : K )

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

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