Let G and H be two groups, , & € Hom (G, H), and A C G be a nonempty subset. (a) Prove ((A)) = ( (A)). Here (A) denotes the subgroup generated by A. (b) Let E = {a € G : (a) = v (a)} . Prove E G.

Q: Let X be a set, G a group, Sx={f:X→X | f is bijective on G} is a group with composition. Given x0∈X…

A: Solution:- given G is a group and X is a subset of G . Here we have to Show subgroup property

Q: Let : G H be a homomorphism. Let (G) = {(g) | ge G}. Show that (G) is a subgroup of H.

A:

Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

A:

Q: 3. Let (G, *) be a group and let H and K be subgroups of G. Prove or disprove each of the following…

A: Given - Let G, * be a group and let H and K be subgroups of G. To find - Prove or disprove each of…

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: Show that a group G = {a ∈  S5; a(1) = 1, a(4) = 4} is a subgroup of S5

A:

Q: Let n > 2 be an integer, and let X C Sn × S, be the set X = {(ơ, T) | 0(1) = T(1)}. Show that X is…

A:

Q: 53. Let H be a subgroup of G and C(H) = {g €G|gh = hg for all he H}. Prove C(H) is a subgroup of G.…

A:

Q: Let in S3 let H = {(1), (12)}. Show that (13)H(23)H ≠ (13)(23)H.(This proves that when H is not a…

A: Given: H = {(1), (12)} We need to prove that (13)H(23)H ≠ (13)(23)H.

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 a e G. Prove that $(a") = ¢(a)" for all n e Z.

A:

Q: 1. a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of (M(R),+)? If yes,…

A: Step 1: Step 2: Step 3: Step 4:

Q: 2. Let G and H be groups, and let a: G H be a homomorphism. Prove the following statements. (a) a(G)…

A: Let us prove above results

Q: Which of the following sets forms a subgroup in the additive group Z? (a) {1} (b) N (c) {1, -1}. (d)…

A:

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

A:

Q: 2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G ×…

A: Step 1: Step 2: Step 3: Step 4:

Q: 1. Let G₁ and G₂ be two groups, with identities e₁ and e2, respectively. Let G = G₁ x G2 and let H…

A:

Q: 1. Let G be a group. Define the center of G by C(G) = {x € G: xg = Prove that C(G) is always a…

A:

Q: ¹) Let H be a subgroup of a group G. Define NĠ (H) : = {g €G | g Hg¯¹ = H}. Prove that N(H) is a…

A:

Q: Q4: Let A and B be subgroups of a group to that A1 = 4 and 1B1-5- Show that AnB = {e}

A: Let A and B be the two subgroups of Group GSuch  thati.e., the Order of subgroup A is 4 and B is 5…