Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system (Zodd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zodd, +) a group? Explain Is (Zodd, +) a commutative group? Explain

Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system (Zodd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zodd, +) a group? Explain Is (Zodd, +) a commutative group? Explain

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 38E: Let n be appositive integer, n1. Prove by induction that the set of transpositions...

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Question

2. Let Z^odd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}.

Determine if the system (Z^odd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative.
Is (Z^odd, +) a group? Explain
Is (Z^odd, +) a commutative group? Explain

Expert Solution

Step 1

The given set is $Z^{o d d} = \{. . . . . . - 5, - 3, - 1, 1, 3, 5, . . . . . .\}$ .

We have to determine

(i) $(Z^{o d d}, +)$ is closed.

(ii) $(Z^{o d d}, +)$ is associative.

(iii) $(Z^{o d d}, +)$ has identity elements.

(iv) $(Z^{o d d}, +)$ has an inverse.

(v) $(Z^{o d d}, +)$ is commutative.

Is $(Z^{o d d}, +)$ group or not.

Is $(Z^{o d d}, +)$ commutative group or not.

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