The socle of a group is the subgroup generated by all minimal normal subgroups. Because the product of normal subgroups is a subgroup, it follows we can remove the word “generated” and replace it by “product.” So the socle of a group is now the product of its minimal normal subgroups. This description can be further refined with a few observations.
If and are minimal normal subgroups then and centralize each other.
Given two distinct minimal normal subgroup and , is contained in and as both are normal. Thus . But and are distinct minimal normal subgroups and is normal so thus . ∎
Let be the socle of . We already know is the product of its minimal normal subgroups, so let us assume where each is a distinct minimal normal subgroup of . Thus and clearly contains and . Now suppose we extend this to a subsquence where
for and for all . Then consider .
As is a minimal normal subgroup and is a normal subgroup, is either contained in or intersects trivially. If is contained in then skip to the next , otherwise set it to be . The result is a squence of minimal normal subgroups where and
As we have already seen distinct minimal normal subgroups centralize each other we conclude that . ∎
If is a minimal normal subgroup of and is characteristic in , then is normal in which contradicts the minimality of . Thus is characteristically simple. ∎
The socle of a finite group is a direct product of simple groups.
As each is characteristically simple each is a direct product of isomorphic simple groups, thus is a direct product simple groups. ∎
|Date of creation||2013-03-22 15:55:12|
|Last modified on||2013-03-22 15:55:12|
|Last modified by||Algeboy (12884)|