Recall that the center of a group G is the set {x € G | xg = gx for all g e G}. Prove that he center of G is a normal subgroup of G.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: a. Prove or Disprove. If H is an abelian normal subgroup of G then H be contained in Z(G).

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Q: (5) Let G be a group. Let H be a subgroup of G and K be a normal subgroup of G. Prove that HK is a…

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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: 2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a…

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

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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: 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 H be a subgroup of a group G. Prove that the set HZG) = {hz | h E H, z E Z(G)} is a subgroup…

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Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

A: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of…

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: 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: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

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Q: Let G be a group and H a subgroup of G. If [G: H] = 2 then H ⊲ G, where [G: H] represents the index…

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Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

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Q: Let H and K be normal subgroups in G such that H n K = {1}. Show that hk = kh for all he H and k e…

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Q: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

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Q: Consider group G and its subgroup H (H ≤ G): [G : H] = 2. Prove why H is a normal subgroup of G.

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Q: 2. Let G = Z₂ x Z₁. (a) Find subgroups H and K of G such that HK but G/H/G/K. (b) Find subgroups A…

A: (a) Here, G = Z2 × Z4                = { (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1,…

Q: Let H be a subgroup of G, define C(H) the centralizer of H.

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