Every group is characteristic in itself

From Groupprops
Revision as of 02:17, 20 February 2013 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., identity-true subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about identity-true subgroup property


Statement

Suppose G is a group. Then, G, viewed as a subgroup of itself, is a characteristic subgroup, i.e., any automorphism of G sends G to within itself.

Related facts

Opposite facts

Note that it is possible for a group to have a subgroup isomorphic to itself that is not characteristic in it. Explicitly, consider a group that is an countable direct power of a nontrivial group. This group is not characteristic as a subgroup of its direct product with itself, even though it is isomorphic to that direct product.

Proof

The proof follows by definition: any automorphism of a group must send it to within itself.