(5) Let G be an arbitrary group. (a) Prove that the formula g*a = ga defines an action of G on itself which is faithful: for any g ≤ G, if g ⋆ a = a for all a EG, then g = 1. (b) Prove that the formula g*a = gag 1 defines an action of G on itself. (c) Find a group G for which the action defined in (b) is not faithful.
(5) Let G be an arbitrary group. (a) Prove that the formula g*a = ga defines an action of G on itself which is faithful: for any g ≤ G, if g ⋆ a = a for all a EG, then g = 1. (b) Prove that the formula g*a = gag 1 defines an action of G on itself. (c) Find a group G for which the action defined in (b) is not faithful.
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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Abstract Algebra
![(5) Let \( G \) be an arbitrary group.
(a) Prove that the formula
\[
g \star a = ga
\]
defines an action of \( G \) on itself which is faithful: for any \( g \in G \), if \( g \star a = a \) for all \( a \in G \), then \( g = 1 \).
(b) Prove that the formula
\[
g \star a = g a g^{-1}
\]
defines an action of \( G \) on itself.
(c) Find a group \( G \) for which the action defined in (b) is not faithful.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F382d69cf-ffe2-43c0-99c4-21bedf550518%2F1c0632d1-1cf7-4548-8123-29a491239e8f%2Ftbjlysm_processed.png&w=3840&q=75)
Transcribed Image Text:(5) Let \( G \) be an arbitrary group.
(a) Prove that the formula
\[
g \star a = ga
\]
defines an action of \( G \) on itself which is faithful: for any \( g \in G \), if \( g \star a = a \) for all \( a \in G \), then \( g = 1 \).
(b) Prove that the formula
\[
g \star a = g a g^{-1}
\]
defines an action of \( G \) on itself.
(c) Find a group \( G \) for which the action defined in (b) is not faithful.
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