Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set is a subgroup of G: {hk | h ∈ H, k ∈ K}}

Q: ) Let G be a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…

A:

Q: 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…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

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: 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: 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 H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

A:

Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: Let H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

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: 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. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

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: Let H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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 group and let H and K subgroup of G. Prove that the intersection H and K is a subgroup of…

A:

Q: Let G be an abelian group. Show that the set containing all elements of G with finite order is a…

A:

Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

A:

Q: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

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: 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: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

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: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

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: 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 H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

A:

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: 7. Let G be a group, and let g E G. Define the centralizer, Z(g), of g in G to be the subset Z(g) =…

A: let G be a group, and let g∈G. Define the centralizer, Zg of g in G to bethe subset…

Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

A:

Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

A:

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

A:

Q: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

A:

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

A: Use the fact that order of subgroup divides order of group

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

A:

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

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: (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: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

A: