Let G be a finite group. The center Z(G) of G is defined to be the set of all group elements z E G such that z.g = g·z for all g € G. Show that Z(G) is an abelian subgroup of G.

Q: Suppose that G is an Abelian group and o is an automorphism of G. Prove that H = {xE G|6(x) = x-1}…

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Q: Let f: G H be an injective homomorphism of groups. Prove G is abelian if H is abelian.

A: Proof given below. Explanation:

Q: Let G be a group. Prove that the mapping a(g) = g-1 for all g in G is an automorphism if and only if…

A: Introduction  A number transformation out of a class onto itself is called a group automorphism.…

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Q: Let G be a group and let a be an element of G. Prove that a func- tion fa : G → G defined by fa(g) =…

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Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: Let G be an Abelian group of order 16. Suppose that there are elementsa and b in G such that |a| =…

A: Given: G be an Abelian group of order 16 and a=b=4.To find: The isomorphism class of G.

Q: Let G be a group. Define ø : G → G by ø(x) = x-1 for all x E G. (a) Prove that ø is one-to-one and…

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Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

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Q: Subgroup of G. (b) Let G be an abelian group. Let T = {ge G: g = e for some positive integer n}.…

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

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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