Definition: Letbe a group wrt the operationand letbe a non-empty subset ofWe say that is a subgroup ofand write ifitself is a group wrt
We can make this more rigorous.
Lemma. Letbe a group wrt the operationand letbe a non-empty subset ofThen is a group wrt ∗ iff
1.for all(i.e.is an operation on),
Proof. If 1 holds thenis an operation onIt must also be associative onsince if the associative law holds in then it must certainly hold for any subset ofBoth 1 and 2 imply thatfor anywhereis the identity element ofwrtso all three axioms of a group are satisfied. The converse is immediate.
Example: The set is a subgroup of wrt complex multiplication. Ifthenso and henceis also a complex 4th root of unity. It follows thatIf thensoFinally, ifthenandIt follows that
Example: The set
is a subgroup of the groupof invertible 2 × 2 matrices with coefficients inIt 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 matrixin S is
Henceis a subgroup of
Example:is a subset ofis a group under the multiplication operation butis not a subgroup since for examplebut 2 has no multiplicative inverse in