A few properties of characteristic subgroups:
Proofs of these properties:
Suppose is the only subgroup of of order . In general, homomorphisms (http://planetmath.org/GroupHomomorphism) take subgroups to subgroups, and of course isomorphisms take subgroups to subgroups of the same order. But since there is only one subgroup of of order , any automorphism must take to , and so .
Take and , and consider the inner automorphisms of (automorphisms of the form for some ). These all preserve , and so are automorphisms of . But any automorphism of preserves , so for any and , .
Let and , and let be an automorphism of . Since , , so , the restriction of to is an automorphism of . Since , so . But is just a restriction of , so . Hence .
|Date of creation||2013-03-22 12:50:56|
|Last modified on||2013-03-22 12:50:56|
|Last modified by||yark (2760)|