Subgroup

The intersection of subgroups is a subgroup

The intersection of any number of subgroups, whether it be countable or uncountable, is itself a subgroup. #m/thm/group For any set of subgroups of a group , the intersection

is itself a subgroup of .

Proof

Clearly for all and therefore . Let . Then for all . This implies that for all and therefore . Therefore is a subgroup by One step subgroup test.

Properties

  1. If each subgroup is a normal subgroup, so too is their intersection.
Proof of 1

Let for , and let . From above, is a subgroup. Now let and . Since each subgroup is normal, for all , hence . Therefore is a normal subgroup.

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