3) We have seen that, for an Abelian group, G, that H= {x=G|x* x=e} forms a subgroup of G. Consider the subset of S: H= (x ES₁ |x*x=e). Recall that the identity for this group is e=p.. 3 (i) Compute H. (ii) Show that H is not a subgroup of S.

Q: Exercise 3.2. Let G, G' be two groups and : G→G' be a (group) homomorphism. (1) Given a subgroup H…

A: Solution-

Q: 1. Let G be an abelian group under multiplication with the identity e. Show that H = {x² | x E G} is…

A:

Q: 2. For both parts below, let p :G → G' be a group homomorphism. Let H' be a subgroup of G'. Show…

A:

Q: Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

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

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Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

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

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Q: Let G be an abelian group, Ha subgroup of G, and n a fixed positive integer. (a) Let Hn = (aEGla"…

A: (a) Given G is an abelian group and H be a subgroup of G. For any fixed n∈ℕ, a subset of G is…

Q: 1. Let G be a cyclic group of order n and k❘n. Show that G has a subgroup of order k; "

A: To prove that a cyclic group ( G ) of order ( n ) has a subgroup of order ( k ) for every ( k ) that…

Q: (True/False) Suppose G is a group of order 25. Then every proper subgroup of G is abelian.…

A: (a) We know any group of order p2 is abelian where p is a prime number. Also, any subgroup of an…

Q: (b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be…

A: We make use of the gcd to answer this question. DEtails are given in step 2.

Q: 4. Let G be a group of order 14. Using Lagrange's theorem, show that every subgroup of G either…

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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)…

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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…

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Q: 2. Let (G, *) and (G', $) be groups, and let : G→ G' be a homomorphism. Let K be a subgroup of G'.…

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Q: (True/False) Suppose G is a group. Then : G x G →G, (x,y) → ay is a group homomorphism. (True/False)…

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Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

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Q: Use the graph shown. f(x)\ 12 8(x) Use the superposition principal to draw a graph of each. Please…

A:

Q: Let G and H be two groups, , & € Hom (G, H), and A C G be a nonempty subset. (a) Prove ((A)) = (…

A: We have proved (a) in two steps, showing both the subset relation one by one. For (b) we have used…

Q: 3. Let G be a group of order pq where p, q are two distinct prime numbers. (a) Assuming that p < q…

A: Given a group G of order pq where p and q are two distinct prime numbers.   Used definition:  number…

Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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Q: 5. Decide if each of the following sets H is a subgroup or not of the group G =R x R, where the…

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Q: 4. Let G be a group and let H, K be subgroups of G such that |H| = 12 and |K| = 5. Prove that HNK =…

A: We have to prove given result: