Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the subgroup generated by aº.

Q: Let G be a group and let H and K be subgroups of G so that H is not contained in K and K is not…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

Q: Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: H and K are subgroup of a group G, then which of the following is a group? a) HUK b) HK c) HnK d)KH.

A: Subgroup: Let (G,*) be a group and H be a non-empty subset of G; then, H is called subgroup if H…

Q: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

Q: Suppose that o: G→G is a group homomorphism. Show that () p(e) = ¢(e') (ii) For every gE G, ($(g))-1…

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Q: Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the…

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Q: Suppose that p:G→G'is a group homomorphism. Show that () p(e) = ¢(e') (1) For every gEG, (ø(g))-l =…

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Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?

A: The expression, G:H can be written as GH .

Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

Q: If d divides the order of a cyclic group then this group has a subgroup of order d. Birini seçin: O…

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Q: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is     a 1   b -1…

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Q: Let a and b belong to a group. If la| = 12, \b| = 22, and (a) N (b) + {e}, prove that a6 = bl1.

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Q: Let G be a group and let a e G. In the special case when A = {a}, we write Cg(a) instead of Cg({a})…

A: The given problem is related with group theory. G is a group and a ∈ G. Here, we write Cga instead…

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

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Q: Suppose that 0:G G is a group homomorphism. Show that () o(e) = 0(e) (i) For every gEG, ($(g))¯1…

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Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

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Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

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

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Q: Let G be a group and let a e G. In the special case when A= {a},we write Cda) instead of CG({a}) for…

A: Consider the provided question, According to you we have to solve only question (3). (3)

Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

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Q: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

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: If H and K are subgroups of G, |H|= 18 and |Kl=30 then a possible value of |HNK| is O18 8. O 4

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Q: Let G be a group and let a e G have order pk for some prime p, where k ≥ 1. Prove that if there is x…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: Although (H,*) and (K,*) are subgroup of a group (G,*) then (H * K, ) may field to be subgroup of…

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Q: . Let G be a group, let g e G, and let H - G. Suppose that the element Hg E G/H has order n. Show…

A: We have to prove that given statement:

Q: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

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Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…

A: According to the given information, let G be a group.

Q: 5. Let H and K be normal subgroups of a group G such that H nK = {1}. Show that hk = kh for all h e…

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Q: Let H and K be two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

A: F hv

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

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Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

Q: Consider the subgroup H = (i) of the the group (C\{0},.). LIT TT TT?

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Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

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

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Q: Let G be a group, and N ⊆ Z(G) be a subgroup of the center of G, Z(G). If G/N, the quotient group is…

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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and A is a subgroup of G,…

Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…

Q: For A, the alternating subgroup of S, show that it is a normal subgroup, write out the cosests, then…

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Q: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

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Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

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Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: Let H and K be subgroups of a group G and assume |G : H| < +co. Show that |K Kn H G H\

A: Let G be a group and let H and k be two subgroup of G.Assume (G: H) is finite.