Examples
The additive group of Z.
The multiplicative groups of Q, R and C.
The multiplicative group of Z/pZ.
Q[X].
Q[X]/(X2+1).
Sn and An.
n by n matrices with nonzero determinant.
The dihedral groups Dn of symmetries of a regular
n-gon.
The invertible elements in a monoid
(Z,+,0, z -> -z) is the additive group of Z.
In Q, R and C every nonzero element has an
inverse with respect to multiplication. So on Q\{0},
R\{0} and C\{0} we have a group structure with
multiplication being the ordinary multiplication.
Suppose p is a prime, then multiplication defines a
group on Z/pZ\{0}.
Indeed, since p is prime, every element has an inverse.
(Q[X],+,0, a -> - a)
is a group. Multiplication does not define a group structure on
Q[X], since X has no inverse.
Let R = Q[X]/(X2+1). Then R
is a field, as the polynomial X2+1
is irreducible. Thus every nonzero element has a multiplicative inverse,
so that the multiplication defines a group on R\{0}.
In Sn and An
every element has an inverse. So both these monoids are also groups.
This of course justifies the names symmetric and alternating group.
Let GLn(R) denote the set of
n by n matrices with real coefficients and nonzero determinant.
Then every
element in GLn(R) has an inverse with respect
to matrix multiplication. Hence GLn(R) is a
group, called the general linear group.
The subset SLn(R) of matrices of determinant 1
also is a group, called the special linear group.
Consider a regular n-gon 
.
A rotation over 2k
/n is a symmetry of
. Also a reflection in a line through
the center and a vertex or the middle of an edge of
is a symmetry.
The n different rotations (including the identity)
and n different reflections form a group denoted by Dn.
.
A rotation over 2k/n is a symmetry of
. Also a reflection in a line through
the center and a vertex or the middle of an edge of
is a symmetry.
The n different rotations (including the identity)
and n different reflections form a group denoted by Dn.
If is a 4-gon then some of the 8
symmetries in D4 are displayed below.
The invertible elements of a monoid M form a group, usually
denoted by M*.