{hk | h ∈ H, k ∈ K}}

Q: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

A:

Q: G. Show that if H and K are subgroups of an abelian group G, then {hk\h E H andke K} is a subgroup…

A:

Q: Let G be a nonabelian group. If H and K are cyclic subgroups of G, does it follow that H ∩ k is also…

A: Given that G is non abelian group. Given that H and K are cyclic subgroups of G. The cyclic group…

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, bEG is either a left or a right coset…

A:

Q: Let ø be a homomorphism from a group G to a group G'. Prove that ker øis a subgroup of G.

A:

Q: Assume that (G, ) is a group and that (H, ) and (K, ) are subgroups of (G,*). Prove that (HnK,*) is…

A:

Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

Q: Let ø : G → G' be an isomorphism of a group (G, *) with a group (G', *'). Prove the following…

A: (A)

Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Prove that the intersection of two subgroups of a group G is a subgroup of G.

A: We will prove the statement.

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

A:

Q: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…

A:

Q: Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

Q: Let G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Given that G is a group and H, K are subgroups of G with the condition that H=<a> and…

Q: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

A:

Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

Q: It is not possible that, for a group G and H and K are nomal subgroups of G, H is isomorphic to K…

A: Let G be a group and H and K are normal subgroups of G

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: Let G be a finite group and let primes p and q ≠ p divide |G|. Prove that if G has precisely one…

A: Given that G be a finite group. Consider G be a finite group and consider prime p and q such that…

Q: Let G be a group with |G|=187 then every proper subgroup of G is:

A:

Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: Prove that if G is a finite cyclic group, Hand K are subgroups of G, and H =f. K, then IHI =f. IKI

A:

Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

A:

Q: Let G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: Let G be a finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

Q: If G is a group and g E G, the centralizer of g E G, is the set CG(g) := {a E G : ag =ga} that is,…

A:

Q: Prove property 1

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A:

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

A:

Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

A:

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Let G be an Abelian group and let H be the subgroup consisting ofall elements of G that have finite…

A:

Q: Show that if G is a finite group with identity e and with an even number of elements, then there is…

A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…

Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

A:

Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

A:

Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

A:

Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: et G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Since H is a cyclic group. and K is cyclic group. H=a,K=b If H⊂K then H∩K=H where H is cyclic If K⊂H…

Q: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

A:

Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: Let G be a group with no proper, nontrivial subgroups and assume that |G| > 1. Prove that G must be…

A: