need help with attached modern algebra. I know that subgroup must be non empty, closed under product and inverses but need help showing it. And showing elements are distinct. Thanks **Problem 8: Group Theory Exploration**Let $ G $ be a group and $ x \in G $.(a) Show that $ H = \{x^n \mid n \in \mathbb{Z}\} $ is a subgroup of $ G $.(b) Show that the elements $ x^n $, where $ n \in \mathbb{Z} $, are all distinct if and only if $ |x| = \infty $.

Answered: Image /qna-images/answer/1107a899-8d8a-42f0-89b7-8c507f9aa859.jpg

Answered: 8. Let G be a group and x E G. (a) Show…

8. Let G be a group and x E G. (a) Show that H {x"|n E Z} is a subgroup of G. (b) Show that the elements x", n E Z, are all distinct if and only if |x| = ∞.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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need help with attached modern algebra. I know that subgroup must be non empty, closed under product and inverses but need help showing it. And showing elements are distinct. Thanks

**Problem 8: Group Theory Exploration**

Let $ G $ be a group and $ x \in G $.

(a) Show that $ H = \{x^n \mid n \in \mathbb{Z}\} $ is a subgroup of $ G $.

(b) Show that the elements $ x^n $, where $ n \in \mathbb{Z} $, are all distinct if and only if $ |x| = \infty $.

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