Question

Let G be a group and $a \in G$. Assume that for every $g \in G$ there exists $k \in \mathbb{Z}^+$ such that $ga = a^k g$. Prove that $H = \langle a \rangle$ is a normal subgroup of G.

Name: Let G be a group and a ∈ G. Assume that for every g ∈ G there exists k ∈ℤ^+ such that ga = a^k g. Prove that H = ⟨ a ⟩ is a normal subgroup of G.
Uploaded: 2023-12-13T11:36:03-08:00
Duration: 4 min 20 s
Channel: Adi S
Description: Let G be a group and a ∈ G. Assume that for every g ∈ G there exists k ∈ℤ^+ such that ga = a^k g. Prove that H = ⟨ a ⟩ is a normal subgroup of G.

          Let G be a group and $a \in G$. Assume that for every $g \in G$ there exists $k \in \mathbb{Z}^+$ such that $ga = a^k g$. Prove that $H = \langle a \rangle$ is a normal subgroup of G.

Added by Pablo J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

12/13/2023

Step 1

To show that H = {a} is a subgroup of G, we need to show that it is non-empty, closed under the group operation, and closed under taking inverses. - Non-empty: Since a is in G, H = {a} is non-empty. - Closure under the group operation: Let x, y be in H = {a}. Then Show more…

Show all steps

Thanks for your feedback!

Please show all relevant work and background Let G be a group and ainG. Assume that for every ginG there exists kinZ^(+)such that ga=a^(k)g. Prove that H=(:a:) is a normal subgroup of G. Let G be a group and a E G. Assume that for every g E G there exists k e Z+ such that ga = ag. Prove that H = (a) is a normal subgroup of G.