Let G be an Abelian group with identity e and let n be some fixed integer.Prove that the set of all elements of G that satisfy the equationxn = e is a subgroup of G. Give an example of a group G in which theset of all elements of G that satisfy the equation x2 = e does not forma subgroup of G.

Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

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Q: Let m ≥ 3. Prove that the symmetric group on m letters is not abelian.

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Q: What is the automorphism group of ?A ut (Z3, +)

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove that  a group of order 1000 cannot be simple

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Q: h(x)=2x^4+5x^3+29x^2+80x-48  Zero:-4i The remaining zeros of h are

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Q: Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian…

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Q: Let us consider that S is subset of group G such that                      S = { x in G ; x^2=e }…

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Q: Let G be an Abelian group. Determine all homomorphisms fromS3 to G.

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Q: Find the non-isomorphic abelian groups of  order 360.

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Q: Show that (up to isomorphism) A5 is the only simple group of order60.

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Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

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Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

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Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

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