Common Groups

The Groups

$D_n$ is the set of all symmetries of a regular n-agon, ie all of its rotations and reflections.
$GL_2(\mathbb{R})$ is the set of invertible $2\times 2$ matrices. The group $GL_2(\mathbb{R})$ is infinite (see the group $SL_2(\mathbb{R})$ – there are an infinite number of matrices with determinant equal to one) and the group operation is matrix multiplication.
$SL_2(\mathbb{R})$ is the set of $2\times 2$ matrices with real number entries, whose determinant is equal to one. There are an infinite number of such matrices, which follows from Bezout’s identity; we may select any two $a,b\in\mathbb{R}$ such that $gcd(a,d) = 1$ . The group operation is composition of functions.
$S_n$ is the set of bijective functions from $\{1, ..., n\}$ to $\{1, ..., n\}$ . The number of elements in $S_n$ is $n!$ . The group operation is composition of functions. We refer to the elements of $S_n$ as permutations.
$O_n$ is the set of distance preserving mappings. For $n=2$ , we end up with the set of symmetries of the circle. The group operation for $O_2$ is composition of rotations and reflections, as rotating and reflecting a circle will not change it’s shape. The order of $O_2$ is infinite.
$SO_2$ is the subset of $O_2$ consisting only of the rotations from $O_2$ .
$\mathbb{Z}^*_m = \{a^{-1} \in \mathbb{Z}_m~|~gcd(m,a)=1\}$ . The condition that $gcd(a,m) = 1$ implies that the inverse of $a$ , $a^{-1}$ , exists, and the operation of the group is multiplication.
The group $\langle \mathbb{R}, + \rangle$ is the set of real numbers with addition. It is infinite.
$\langle \mathbb{Z}, + \rangle$ is a group. It is infinite, and uses the addition operations.
$\langle \mathbb{C}, + \rangle$ is another infinite group with the addition operation.
$\langle \mathbb{Q}, + \rangle$ is another infinite group, which uses the addition operation.
$\langle \mathbb{Z}_n, + \rangle$ is a group with $n$ elements. It uses the addition operation modulo $n$ , and it’s elements are $\langle \mathbb{Z}_n, + \rangle = \{\overline{0}, \overline{1}, ..., \overline{(n-1)}\}$ .
$\langle \mathbb{R}^*, \times \rangle$ is a subset of real numbers with multiplication, and $\mathbb{R}^* = \mathbb{R} \setminus \{0\}$ . Clearly the order of the group is infinite. We exclude the element $0\in\mathbb{R}$ because $0$ has no inverse under multiplication.
$\langle \mathbb{Q}^*, \times \rangle$ the group of non-zero rational numbers, combined with multiplication. Ie, $\mathbb{Q^*} = \mathbb{Q}\setminus \{0\}$ . We exclude the element zero for the same reason as for $\langle \mathbb{R}^*, \times \rangle$ : because $0$ has no inverse under multiplication. Clearly the group is infinite.
$A_n$ is the subgroup of $S_n$ with the condition that elements of $A_n$ are the elements of $S_n$ that can be written as an even number of transpositions. The group has $\frac{n!}{2}$ elements. We refer to this group as the alternating subgroup.