Let H be a subgroup of a group G and a, b € G. Then b E aH if and only it O None of these O ab e H O ab-1 eH a-1b e H

Q: 1. Let G be a group and H a nonempty subset of G. Then H <G if ab-EH whenever a,bEH

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Q: Let G1 and G2 be groups, and let G = G1 × G2. Write e and e2 for the identities of G1 and G2…

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Q: Let M be a subgroup of group G, and a,b e G, then aM=bM→ a-1 b € M True O False O

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Q: Let H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

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: 3) Prove that if A and B are subsets of G with A C B then Cc(B) is a subgroup of CG(A).

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Q: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…

A: We know that  b∈bH                    (1) We know that  aH = bH if and only if a-1b ∈H…

Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

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Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy&lt;minox, oy. Let us take the group…

Q: Let G be a group. Let x EG be such that O(x) = 4. Then: * O (x^12) = 5 O O(x^15) = 5 O O(x^10) = 5…

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

Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

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Q: 10. Let G be a group and H a subgroup of G. Let CH) - {gEG|gh = hg. VhEH}, Show that C(H) <G.

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Q: If G is a group then Z(G)◄G. *True False

A: Use the relevant definitions.

Q: (b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.

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Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: Let H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba EH O None of these…

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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: 9) Let H be a subgroup of a group G and a, b E G. Then a E bH if and only if O b-1a E H O None of…

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Q: Let H be a subgroup of a group G and a, bEG. Then b e aH if and only if* O ab-1 eH O ab e H O None…

A: See the detailed solution below.

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if* O ba e H O b-1a e H…

A: We will use definition of left coset

Q: If H and K are subgroups of the group G, then which one of the following is also a subgroup of G? a.…

A: Answer

Q: Let H be a subgroup of a group G and a, b EG. Then b E aH if and only if O None of these O ab EH О…

A: The solution is :

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: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…

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Q: If H and K are subgroups of the group G, then which one of the following is also a subgroup of G? O…

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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…

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Q: An element a of a group G has order n E z+ if and only if a" = e.

A: The given statement is False.

Q: 9) Let H be a subgroup of a group G and a, be G. Then a E bH if and only it O ba-1 eH O ba eH O b-1a…

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Q: Let H be a subgroup of a group G and a, bEG. Then bE aH if and only if * O None of these O ab e H O…

A: here option (c) is true.

Q: Let G be a group and a be an element of this group such that a^63e. The possible orders of a are: *…

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Q: Let G be a group and a be an element of this group then necessarily: * O lal=|G| |a|</G| O lals|G] O…

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Q: Let G be a group and a 2 G. Then the set = {a" : ne Z}is a subgroup of G. 48.

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Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O None of these O b-1a 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: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba-1 E H ba E H O b-1a…

A: Q9. Third option is correct.

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: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

Q: Let (G,*) be any group and (a) = {a'| i = 0, +1, F2, F: (a) = {... , a-2, a-1, a° = e, %3D %3D…

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Q: let G be a group, a,b E G  such that bab^-1 =a^r , for some  r E N, where N are the natural ones,…

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Q: 9) Let H be a subgroup of a group G and a, bEG. Then a e bH if and only if* O ba e H O None of these…

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Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * None of these b-1a e H…

A: Second option is correct.

Q: Let G be a group and a be an element of this group such that a12 = e. The possible orders of a are:…

A: Option (4)

Q: Let K and H be subgroups of a finite group G with KCHCG.lf [G:K] = 12 and [H:K] = 3. Then, [G:H] =…

A: Let , K  and  H  be subgroups of finite group G. Also . K ⊆ H ⊆ G Here , G : K = 12  , H : K = 3 We…

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 be a subgroup of G and let a, be G. If Ha Hb, then* %3D aH = bH O a-1H = b-1H O Ha = Hb Ha-1 =…

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Q: Let G be a group. Let x, y e G be such that O(x) = 7, O(y) = 2, x^6 y = yx. Then O(xy) is O Infinity…

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Q: Let H be a subgroup of a group G and a, bEG. ThenbE aH if and only if * O None of these O ab EH O…

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