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: 1. Give, if possible, one generator for the subgroup H = of Z. Justify your answer.

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Q: Suppose the Cayley table for B = {e, a, b, c}unde the binary operation * is given by * е а е е а b a…

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Q: Let f be the following permutation: 1 2 3 4 5 6 7 f = 4 3 6 2 1 5 7 a. Determine the group to which…

A: Given permutation is f=12345674362157 f is permutation of 7 elements so it is belong to s7. i.e. sn…

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 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: Let K be a group with 8 elements. Let H be a subgroup of K and H<K. It is known that the order of H…

A: Given:- K is a group with 8 elements , H is a subgroup of K and H&lt;K. H has order of atleast 3.…

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

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Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

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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: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

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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] = 3 4…

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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: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

Q: 27. 28. Define and give an example of a cyclic group. Is every group a cyclic? Why?

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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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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: nite, or Abelian,

A: Given that G is a group, not necessarily cyclic, finite, or Abelian. Let x∈G. The order of the…

Q: let n be a fixed natural number.Verify that the set  n.Z={n.k l kEZ} is a group of (Z,+).

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Q: In Z, find all generators of the subgroup &lt;m&gt; &lt;n&gt;.

A: The objective of the question is to find all generators of the subgroup &lt;m&gt; &lt;n&gt; 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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