## Subgroups

Definition: Let be a group wrt the operation and let be a non-empty subset of We say that is a subgroup of and write if itself is a group wrt We can make this more rigorous.

Lemma. Let be a group wrt the operation and let be a non-empty subset of Then is a  group wrt ∗ iff

1. for all (i.e. is an operation on ),

2. for every Proof. If 1 holds then is an operation on It must also be associative on since if the associative law holds in then it must certainly hold for any subset of Both 1 and 2 imply that for any where is the identity element of wrt so all three axioms of a group are satisfied. The converse is immediate.

Example: The set is a subgroup of wrt complex multiplication. If then so and hence is also a complex 4th root of unity. It follows that If then so Finally, if then and It follows that Example: The set is a subgroup of the group of invertible 2 × 2 matrices with coefficients in It is clearly a subset. The product of any pair of invertible upper triangular matrices is also invertible and upper triangular, and  the inverse of any matrix in S is Hence is a subgroup of Example: is a subset of is a group under the multiplication operation but is not a subgroup since for example but 2 has no multiplicative inverse in  