In mathematics, a subgroup of a group is fully characteristic (or fully invariant) if it is invariant under every endomorphism of the group. That is, any endomorphism of the group takes elements of the subgroup to elements of the subgroup.
Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. Every fully characteristic subgroup is a strictly characteristic subgroup, and a fortiori a characteristic subgroup.
The commutator subgroup of a group is always a fully characteristic subgroup. More generally, any verbal subgroup is always fully characteristic. For any reduced free group, and, in particular, for any free group, the converse also holds — every fully characteristic subgroup is verbal.
See also characteristic subgroup.
Famous quotes containing the word fully:
“I allude to these facts to show that, so far from the Supper being a tradition in which men are fully agreed, there has always been the widest room for difference of opinion upon this particular. Having recently given particular attention to this subject, I was led to the conclusion that Jesus did not intend to establish an institution for perpetual observance when he ate the Passover with his disciples; and further, to the opinion that it is not expedient to celebrate it as we do.”
—Ralph Waldo Emerson (1803–1882)