Multiplicative group modulo n
From Groupprops
Definition
Let be a positive integer. The multiplicative group modulo
is the subgroup of the multiplicative monoid modulo n comprising the elements that have inverses.
Equivalently, it is the group, under multiplication, of elements in that are relatively prime to
. (The two definitions are equivalent because if
and
are relatively prime, there exist integers
such that
, so
).
Facts
- The order of the multiplicative group modulo
equals the number of elements in
that are relatively prime to
. This number is termed the Euler-phi function or Euler totient function of
, and is denoted
.
- For a prime
,
. In other words, every nonzero element less than
is invertible modulo
.
- The multiplicative group modulo
is a cyclic group if and only if
for
an odd prime and
a natural number.