O If a group G acts on a set S, every element of S is fixed by the identity of G. O Every group of order 42 has a normal subgroup of order 8. Qx Q is a cyclic group.

Q: One of the following is True O If aH = bH, then Ha = Hb. O Only subgroups of finite groups can have…

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Q: Let m be a positive integer. If m is not a prime, prove that the set {1,2,..., m – 1} is not a group…

A: We show that it doesn't satisfy clousre property.

Q: . Let Sn be the symmetric group on n elements, let An Sn be the alternating subgroup, consisting of…

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Q: G is a group with identity element e.  If a and b are elements of G, and n is any integer, show…

A: Let if n=0, the statement 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: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

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Q: (c) Prove that for every divisor d of n, Zn has a unique subgroup of order d. (Hint: What is the…

A: C) Let k be a subgroup of order d, then k is cyclic and generated by an element of order k =K⊂H…

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: In Z, find all generators of the subgroup <m> <n>.

A: The objective of the question is to find all generators of the subgroup <m> <n> in the…

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 G be a group of order 21. Show that the number of order 3 subgroups of G is either one or…

A: Given that let G be a group of order .To prove:- The number of order  subgroups of G is either one…

Q: 3. Show that if G is a cyclic group then every subgroup of G is cyclic.

A: Subgroup of a cyclic group

Q: Question 2: a. Let G be a group and a E G. Show that if |a| = n, then |a¹| = n. b. Without computing…

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