Let
be a group. Show that a nonempty subset
which is closed under
and inversion is a group under the restriction
of
to
.
We need to demonstrate that
is a binary operator on
, that
is associative, that an identity element
exists, and that every element has an inverse.
is indeed a group under the restriction of
.
We need to demonstrate that
is a binary operator on
by hypothesis.
is associative because
is associative.
- Since
is nonempty, there exists some
. Since
is closed under inversion in
,
. And since
is closed under
, we have
. Moreover, for all
we have
; thus
has an identity element and in fact
.
- Let
. Then since
is closed under inversion in
we have
, and
. Thus every element of
has an inverse.
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