Question

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 )

Answer #1

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

(a) Prove or disprove: if H and K are subgroups of G, then H ∩ K
is a subgroup of G.
(b) Prove or disprove: if H is an abelian subgroup of G, then G
is abelian

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 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.

(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.

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

(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.

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 5 minutes ago

asked 11 minutes ago

asked 19 minutes ago

asked 24 minutes ago

asked 24 minutes ago

asked 33 minutes ago

asked 33 minutes ago

asked 40 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago