2 Suppase G is an abeli Group of Order 6 Contauining an element' of order 3. Prove G is Cyclic. 2 Suppase G has only one element (Say a) of order 2. Show xa = ax 'for all % for all in G.

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

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Q: Find two elements of maximum order in the group G = Z100 Z4 O Z2. How many such elements are there?…

A: As Z100 has no element of order 100 otherwise it will be cyclic which is not true. So, we consider…

Q: 4. Let (A,) be a monoid such that for every x in A, x *x = e, where e is the identity element. Show…

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Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

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Q: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: Let G be a group and a ∈ G. Suppose that |a| = 18, find |a^21|. Suppose that |a| = 32, find a…

A: We are given that G is a group, and a ∈ G.A) The order of element a is given as 18. a=18 ⇒a18 = e,…

Q: 16. True or False: (a). Every element in a cyclic group (G) is a generator of G. (b). A group (G, *)…

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Q: for the cyclic group of order 16 generated by a , show that a^4 generates an invariant subgroup

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Q: The group (U20, X20) = {(1, 3, 7, 9, 11, 13, 17, 19), ×20} has subgroups H = {1,3,7,9) and K =…

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Q: Express the following as products of cyclic gr a) Z140 ») U (140) =) Z400 H) U (400).

A: To express the given groups as products of cyclic groups, we need to find the prime factorization of…

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: (a) (Z,*) with a × b = a/b is (b) (Q, *) with a * b = ab - a - b is (c) (Q,*) with a*b = a² +6² is…

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Q: 8. Prove that Q is not a subcartesian product of infinite cyclic groups.

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Q: 2- Find the cyclic subgroups in (Z,, +,) generated by 3,5.

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Q: Thus option (c) is correct. Let G be a group with G| = 96. Suppose H and K are subgroups of G with…

A: for abelian group -: condition -: 1 closure ,2 associative , 3 -: identity ,  4 -: inverse ,5-:…

Q: ^^)^^\Y) 13. If * is defined on Z such that a*b = a + b -1 for a, b E Z, show that {Z,*} is an…

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Q: Prove that (Z × Z)/((0,1)) is an infinite cyclic group. Prove that (Z × Z)/((1,1)) is an infinite…

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Q: (b) Define Dn → Z₂ by 0(p) = 0 and 0(op') = 1. Show that is a group ho- momorphism. State its kernel…

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Q: . Prove that the order of any subgroup of a finite group divides the order of the group.

A: We have to prove the Lagrange 's theorem.