Group Theory Question: Is (Left) Multiplication by g an Isomorphism in G?

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Homework Help Overview

The discussion revolves around the properties of group operations, specifically examining whether left multiplication by an element g in a group G constitutes a bijection from G to itself.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the definition of isomorphism versus bijection, with some attempting to clarify the conditions under which left multiplication by g is a bijection.

Discussion Status

Some participants express confidence in the assertion that left multiplication by g is a bijection, while others emphasize the need to demonstrate this property through a direct approach. There is a mix of agreement and further exploration of the reasoning involved.

Contextual Notes

There is a noted confusion between the terms isomorphism and bijection, which influences the clarity of the discussion. Participants are encouraged to refine their understanding of these concepts as they relate to group theory.

ehrenfest
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Homework Statement


true or false:
If G is a group and g is in G. Then (left) multiplication by g is an isomorphism from G to G

Homework Equations


The Attempt at a Solution


I am pretty sure it is true since ax=b always has a solution if a and b are in group. But can someone just confirm this?

EDIT: sorry, I don't mean isomorphism, I mean bijection
 
Last edited:
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Yes, it's true.
 
ehrenfest said:

Homework Statement


I am pretty sure it is true since ax=b always has a solution if a and b are in group. But can someone just confirm this?

EDIT: sorry, I don't mean isomorphism, I mean bijection

it's true as nate said, but your reason doesn't tell the whole story(it just gives surjectivity). Try to do it directly, for a fixed g in G, define phi:G->G by phi(x) = gx. Now show it's a bijection.
 
gx is in g, and if g1x=g2x then g2inv g1 x = x so g1=g2 so all elements are different, so you can just make couple in your head from every g to every gx.

I passed group theory this monday =D
 

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