Q: Q25: What's mean by a zero divisor element of a ring R? Find all zero divisor elements of Zg-
A: see below the answer
Q: a. Find the elements of Zg that have multiplicative inverses.
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A: Given
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A: Explanation of the answer is as follows
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Q: zzing/user/attempt/quiz_start_frame_auto.d21?ou%3D184337&isprv3&drc3D0&qi3D249474&cfgl%=D0&dnb%3D0&f…
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Q: 33) p
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- Let G be a group of finite order n. Prove that an=e for all a in G.9. Let be a group of all nonzero real numbers under multiplication. Find a subset of that is closed under multiplication but is not a subgroup of .Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.Exercises 12. Prove that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. (Hint: Consider the mapping defined by for all .)44. Let be a subgroup of a group .For, define the relation by if and only if . Prove that is an equivalence relation on . Let . Find , the equivalence class containing .
- 6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .Let G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.
- Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Sec. 3.1,52 Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations.Exercises 35. Prove that any two groups of order are isomorphic.Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exist an aG such that y=a1xa, is an equivalence relation. Let xG. Find [ x ], the equivalence class containing x, if G is abelian. (Sec 3.3,23) Sec. 3.3, #23: 23. Let R be the equivalence relation on G defined by xRy if and only if there exists an element a in G such that y=a1xa. If x(G), find [ x ], the equivalence class containing x.