The intersection of subgroups is a subgroup
The intersection of any number of subgroups, whether it be countable or uncountable, is itself a subgroup. group
For any set
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
- 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.