Suppose G is an abelian group and let S = {a € G|a = x² for some x = G} be the set of squares in G. Prove that S is a subgroup of G.

Q: SUCH THAT LET H BE A PROPER SUBGROUP OF G V x,y € G-H, xy EH. PROVE THAT HAG.

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Q: Let G be an abelian group and let H= {g EG: g^n= e for some nEN}. Prove that H is a subgroup of G.

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Q: Suppose that H is a subgroup of G and let G act by conjugation on the set X = {gHg-1 : g ∈ G}.…

A: Given that,H is a subgroup of G and G act by conjunction on the set

Q: Prove that a simple group cannot have a subgroup of index 4.29. Prove that there is no simple group…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Let G be an abelian group with identity e. Prove that is a subgroup of G. H = {g E G|g" = e for some…

A: G is an abelian group and H is a subset 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 an abelian group and m be a  positive integer.  Then the set             mG={mn∣x∈G}  and…

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Q: If f is a homomorphism of a group G into a group G' with Kernal K. Let a € 6 be such that f(a) = a'…

A: We are given that f : G → G' is a homomorphism. Kernel of f is K.     => for all elements x in K,…

Q: Let G be a finite group of order n . Let g be an element of G with order k. Prove that k|n

A: To prove ''Let G be a finite group of order n . Let g be an element of G with order k. Prove that…

Q: Let H be a normal subgroup of G and K a subgroup of G that containsH. Prove that K is normal in G if…

A: Assume that H is a normal subgroup of G, and that K is a subgroup of G containing H. Then, we need…

Q: Prove that if H is a normal subgroup of G st H and H/G are finitely so is G.

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Q: If H and K are subgroups of G and g belongs to G, show that g(H N K) = gH n gK.

A: Given: H and K are subgroups of G and g.To prove: gH∩K=gH∩gK

Q: If H is a finite subgroup of a group G and x ∈ G, prove that |N(H)| = |N(xHx-1)|.

A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

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 a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

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 G and H be groups, and let N be a normal subgroup of G. Decide whether each of the following two…

A: (a) The statement "The group  is abelian if and only if G and H are both abelian" is…

Q: Assume that G is an abelian group. Let H C G be a subgroup. Prove that the factor group G/H is…

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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: Let H be a subgroup of a group G. Prove that the number of conjugates of H in G is |G:N(H)|.

A: It is given that H be a subgroup of a group G. It is required to prove that the number of conjugates…

Q: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

Q: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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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 an abelian group. prove: H = {g € G||g| <∞o} is a subgroup of G. We need to prove that H ‡…

A: The solution effectively utilizes the theorem that states if x and y are elements of a subset H,…

Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

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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…

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Q: Let G be a finite group and H1, H2, ., Hk be subgroups of G. (a) Show that k H1n H2n. n Hk < G. Hi…

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Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

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Q: (2) Let G be a finite group and H and K be subgroups of G such that HCKCG. Prove that [G: H] = [G:…

A: Let us prove this beautiful result using Lagrange's theorem. Lagrange's theorem = If G is a finite…

Q: Let : G → H be a group homomorphism. Show that is a subgroup of H. Im(6) := {p(g) | g = G}

A: Let G,∘ and H,* be two group and ϕ:G→H be a homomorphism. Let ϕG is a non-empty subset of H. Since…

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

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Q: prove that Uk(n) is a subgroup of U(n)

A: The objective of this question is to prove that Uk(n) is a subgroup of U(n). Here, U(n) represents…