Subgroups
Definition: Let
be a group wrt the operation
and let
be a non-empty subset of
We say that is a subgroup ofand write
ifitself is a group wrt
We can make this more rigorous.
Lemma. Letbe a group wrt the operation
and letbe a non-empty subset ofThen is a group wrt ∗ iff
1.
for all
(i.e.is an operation on),
2.
for every
Proof. If 1 holds thenis an operation on
It must also be associative onsince if the associative law holds in then it must certainly hold for any subset ofBoth 1 and 2 imply that
for any
where
is the identity element ofwrt
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, ifthen
and
It follows that
Example: The set
is a subgroup of the groupof 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 butis not a subgroup since for example
but 2 has no multiplicative inverse in
