Question

Prove that no group can have exactly two elements of order 2.
Suppose that
a, b
∈
G
are two distinct elements of order 2 in a group
G
. Note that because
a
6
=
b
, it cannot be the case that
ab
=
e
. By cancellation,
ab
=
a
and
ab
=
b
are also
ruled out. If
ab
=
ba
, then (
ab
)
2
=
abab
=
aabb
=
e
and
ab
is a third element of order
two. If
ab
6
=
ba
, than an immediate consequence is that
aba
6
=
b
, and we also know that
aba
6
=
e
and
aba
6
=
a
because
ba
6
=
a
and
ba
6
=
e
. Thus
aba
is distinct from
e
,
a
, and