- #1

wheezyg

- 5

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

In summary, the conversation discusses the concept of Aut(G) being a subgroup of G and the distinction between inner and outer automorphisms. The speaker is seeking clarification and recommendations for further reading on this topic.

- #1

wheezyg

- 5

- 0

Physics news on Phys.org

- #2

boboYO

- 106

- 0

- #3

arkajad

- 1,481

- 4

[tex]\rho_g(h)=ghg^{-1}[/tex]

Thus [tex]G[/tex] has an image, possibly with a non-trivial kernel, in [tex]\mbox{Aut}(G)[/tex] - these are called "inner automorphisms". But, in general, there can be also "outer automorphisms" - automorphisms of G that can not be implemented by any element of G.

A group automorphism is a function that maps a group onto itself while preserving the group operation. This means that the elements of the group are rearranged in a way that the group operation remains unchanged.

A subgroup is a subset of a group that also forms a group under the same operation. A group automorphism is a function that preserves the group operation, but it does not necessarily form a subgroup. In other words, a group automorphism is a rearrangement of the elements in a group, while a subgroup is a subset of the original group.

A group automorphism is not always a subgroup because it may not satisfy the closure property, which is necessary for a subset to be a subgroup. This means that the result of applying the automorphism to two elements in the group may not always be an element of the subset.

No, a group automorphism cannot be a proper subgroup. A proper subgroup is a subset of a group that is also a subgroup, meaning that it contains all the necessary elements for the operation to be closed. However, a group automorphism only rearranges the elements in a group and does not necessarily form a subgroup.

An example of a group automorphism that is not a subgroup is the identity automorphism, which maps each element in a group to itself. This is because it does not change the group operation, but it does not form a subgroup because it does not satisfy the closure property.

- Replies
- 5

- Views
- 2K

- Replies
- 2

- Views
- 1K

- Replies
- 2

- Views
- 1K

- Replies
- 3

- Views
- 2K

- Replies
- 3

- Views
- 1K

- Replies
- 10

- Views
- 2K

- Replies
- 6

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Replies
- 1

- Views
- 1K

- Replies
- 1

- Views
- 1K

Share: