Let
and for
let
be the fractional part of
(i.e.,
). Prove that
is a well-defined binary operation on
and that
is an abelian group under
.
Before we begin, note that
for all
and
as follows. For all integers
, we have
iff
iff
iff
.
is a group under
.
Before we begin, note that
- We first show that
is well defined. Suppose
. There are two cases for
. If
, then we have
and so
. If
, we have
. But since
,
. Thus
. So
is indeed a binary operator on
.
- We now show that
is associative. To that end, let
. Then we have the following.
= = = = = = = = .
- We now show that
is the identity element of
under
. If
, then
. Thus we have
and
.
- We now show that every element has an inverse. If
, then
. Then
and similarly
. Certainly, if
then
.
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