Question

12. Let G be a group. Prove that the mapping $\alpha(g) = g^{-1}$ for all g in G is an automorphism if and only if G is Abelian. 13. If g and h are elements from a group, prove that $\phi_{g}\phi_{h} = \phi_{gh}$.

Name: 12 let g be a group prove that the mapping alphag g 1 for all g in g is an automorphism if and only if g is abelian 13 if g and h are elements from a group prove that phigphih phigh 38682
Uploaded: 2023-04-07T15:31:34-08:00
Duration: 2 min 32 s
Channel: Sri K
Description: 12 let g be a group prove that the mapping alphag g 1 for all g in g is an automorphism if and only if g is abelian 13 if g and h are elements from a group prove that phigphih phigh 38682

          12. Let G be a group. Prove that the mapping $\alpha(g) = g^{-1}$ for all g in G is an automorphism if and only if G is Abelian.
13. If g and h are elements from a group, prove that $\phi_{g}\phi_{h} = \phi_{gh}$.

12 let g be a group prove that the mapping alphag g 1 for all g in g is an automorphism if and only if g is abelian 13 if g and h are elements from a group prove that phigphih phigh 38682

Added by John C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/07/2023

Step 1

First, let's recall the definition of an automorphism. An automorphism is an isomorphism from a group to itself. An isomorphism is a homomorphism that is also a bijection (one-to-one and onto). So, we need to prove two things: 1. If $\alpha$ is an automorphism, Show more…

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12. Let G be a group. Prove that the mapping $\alpha(g) = g^{-1}$ for all g in G is an automorphism if and only if G is Abelian. 13. If g and h are elements from a group, prove that $\phi_{g}\phi_{h} = \phi_{gh}$.