3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b. An element g e G is said to be square if there exists x € G such that g = x2. Prove that G has odd order if and only if every element of g is a square. |

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

Q: Exercise 3.2.6 Show that if G and H are isomorphic groups, then G commutative implies H is…

A: A group G is called Commutative if for any a,b in G imply ab=ba

Q: 9.12 Let G be a group and for elements a, b e G let a R b mean that there exists an element x = G…

A:

Q: 3*. Let (F,+) be the group of polynomials over3 X {x, a2, a3,...} be the set of all positive integer…

A: To describe the subgroup generated by the positive powers of x in the additive group F[x]

Q: 3. Let G be a group. Recall that if x E G, we defined the conjugacy class of x in G, denoted by…

A:

Q: 2. Let G be a group. Recall we showed in class that if |G| is finite, then o(g) is finite for all g…

A: be a group.If  is finite then  is also finite where We have to show the converse of the statement…

Q: 7. Let G be a group and for elements a, b e G define aRb if and only if there exists an element x E…

A:

Q: Suppose that Z10 and Z15 are both homomorphic images of a finitegroup G. What can be said about |G|?…

A: Given: Z10 and Z15 are both homomorphic images of a finite group G. We have to tell about G and…

Q: Let a and b be elements of a group G. We define the conjugacy relation a-b by the following…

A: According to the given information,

Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

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: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

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

A:

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: 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: Let G and H be finite groups. Prove the following statements. (a) If a: G H is a homomorphism, then…

A: Given that  and  are finite groups.We need to prove the following.(a) If  is a homomorphism, then…

Q: 1. Let G be a p-group acting on a finite set X. Let XG be the set of fixed points of this action: XG…

A: The solution lies using the Orbit Stabilizer Theorem

Q: 5 Let G be the set of all functions f: R → R of the form f(x) = ax + b where a and b are real…

A: Introduction: Function composition is an operation where two functions are combined to form a new…

Q: Exercise 7.15. Let G be a finite group, g € G, and N≤ G. Prove that the order of gN in G/N must…

A:

Q: 3. Let G be a finite group of order |G| = n. Prove that G is isomorphic to a subgroup of Sym(G)…

A: The objective is to prove that G is isomorphic to a subgroup of symmetric group.

Q: A group G acts faithfully on a set S if g * s = s for all s € S implies g = e. Show that G acts…

A: Proof 1: If G acts faithfully on S then mg≠mg', where g≠g' ∈G. Let G acts faithfully on S. Assume to…

Q: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: If G and H are groups with identities 16 and ¹H, respectively, and f:G ---> H is a homomorphism,…

A:

Q: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: Prove Corollary 36.4 which says, "Let G be a finite group. Then G is a p-group if and only if |G| is…

A: First suppose that G is a p-group. Let q is a prime which divides |G|, then G contains an element of…