Question

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, provided that H and K are subgroups of G. Remember that what needs to be done to establish that some element x belongs to H ∩ K is to show that x belongs to H and x belongs to K. Your answer to part (b) should present a specific group with a pair of specific subgroups, whose union is not a group for some demonstrated reason. Examples of pairs of subgroups like this are plentiful, once you give yourself a particular group in which to work.

Answer #1

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

(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.

(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 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

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.

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.

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 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].

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