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 ab-1 e H ab 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 G be a group and a be an element of this group then necessarily: * Jal<|G| O lal2|G| O lal=|G|

A: In the given question we have to chose the correct option from the given options of the given…

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 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 let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

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: 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: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…

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Q: Let G be a group. Show that Z(G) = nEC(a). [This means the intersection of all subgroups of the form…

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Q: that a and xax^-1 have same order for all x belongs G.

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Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…

A: First option is correct.

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: V Let G be a group. Show that for all a, b in G, (a*b) =b*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 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 A and B be subgroups of G such that AB = BA. Then AB is a subgroup of G.. Select one: True False

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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: Let H be a subgroup of G. If a and b are elements of G such that aH = bH, then Ja| = |b|. %3D

A: Given that Let H be a subgroup  of G. If a and b are elements of G such that aH = bH, then a=b. To…

Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…

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Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

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: 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: 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: Let G be a group and a be an element of this group : then necessarily O laisIGI lal2/G] O lal=IG]

A: Given ,  Let G be a group and a be an element of this group

Q: Let a,b be elements of a group G .Assume that a has order 5 and  a^3b=ba^3. prove that ab=ba.

A: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.

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, b € G. Then b E aH if and only it O None of these O ab e H 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 * O None of these O ab e H O…

A: here option (c) is true.

Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

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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: The set A := {(1), (1 2 3), (2 3 4)} forms a subgroup of the permutation group (G,.). O True O False

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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: 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: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: O…

A:

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: 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…

A:

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 a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…

A: Result: Let G be a group and H be a normal subgroup of G. Let 'a' be an element of G such that order…

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 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 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…

A: