Binary operation
A binary operation \(\star\) on a set \(G\) is a function \(\star: G \times G \to G\). For any \(a,b \in G\) we shall write \(a \star b\) for \(\star(a, b)\).
Group
A group is an ordered pair \((G, \star)\) where \(G\) is a set and \(\star\) is a binary operation on \(G\) satisfying the following axioms:
- \(\star\) is associative: \((a \star b) \star c = a \star (b \star c)\) for all \(a, b, c \in G\)
- there exists an identity \(e\) in \(G\) such that for all \(a \in G\), \(a \star e = e \star a = a\)
- for each \(a \in G\), there is an inverse \(a^{-1} \in G\) of \(a\) such that \(a \star a^{-1} = a^{-1} \star a = e\)
Abelian group
The group \((G,\star)\) is called abelian (or commutative) if \(a \star b = b \star a\) for all \(a,b \in G\).
Order
For \(G\) a group and \(x \in G\), define the order of \(x\) to be the smallest positive integer \(n\) such that \(x^n = 1\), and denote this integer by \(|x|\). In this case, \(x\) is said to be of order \(n\). If no positive power of \(x\) is the identity, the order of \(x\) is defined to be infinity and \(x\) is said to be of infinite order.
Group table
Let \(G = \{g_1, g_2,…,g_n\}\) be a finite group with \(g_1 = 1\). The multiplication table or group table of \(G\) is the \(n \times n\) matrix whose \(i,j\) entry is the group element \(g_ig_j\).
Example
Symmetric group
Let \(\Omega\) be any nonempty set and let \(S_\Omega\) be the set of all bijections from \(\Omega\) to itself. In other words, \(S_\Omega\) is the set of all permutations of \(\Omega\). \(S_\Omega\) is a group under function composition \(\circ\). This group is called the symmetric group on the set \(\Omega\).
In the special case where \(\Omega = \{1,2,…,n\}\), the symmetric group on \(\Omega\) is denoted \(S_n\), the symmetric group of degree \(n\).
Homomorphism
Let \((G, \star)\) and \((H, \diamond)\) be groups. A map \(\varphi:G \rightarrow H\) such that $$ \varphi(x \star y) = \varphi(x)\diamond\varphi(y),\text{ for all } x, y \in G $$ is called a homomorphism.
Isomorphism
The map \(\varphi:G \rightarrow H\) is called an isomorphism and \(G\) and \(H\) are said to be isomorphic or of the same isomorphism type, written \(G \cong H\), if
- \(\varphi\) is a homomorphism, and
- \(\varphi\) is a bijection.
Group action
A group action of a group \(G\) on a set \(A\) is a map from \(G \times A\) to \(A\) (written as \(g \cdot a\) for all \(g \in G\) and \(a \in A\)), satisfying the following properties.
- \(g_1 \cdot (g_2 \cdot a) = (g_1g_2) \cdot a\) for all \(g_1, g_2 \in G, a \in A\), and
- \(1\cdot a = a\) for all \(a \in A\).
We say \(G\) is a group acting on a set \(A\).
Facts
Let the group \(G\) act on the set \(A\). For each \(g \in G\), define \(\sigma_g : A \rightarrow A\) by \(\sigma_g(a) = g\cdot a\). Then,
- for each fixed \(g \in G\), \(\sigma_g\) is a permutation of \(A\), and
- the map from \(G\) to \(S_A\) defined by \(g \mapsto \sigma_g\) is a homomorphism.
Proof
For an arbitrary \(g \in G\),
$$ (\sigma_{g^{-1}} \circ \sigma_g)(a) = \sigma_{g^{-1}}(\sigma_g(a)) = \sigma_{g^{-1}}(g\cdot a) = g^{-1} \cdot (g\cdot a) = (g^{-1}\cdot g)\cdot a = 1 \cdot a = a. $$
The same holds when \(g\) and \(g^{-1}\) are interchanged. Therefore, \(\sigma_g\) has a 2-sided inverse, and thus is a permutation.
Define \(\varphi\) as \(\varphi(g) = \sigma_g\).
$$ \varphi(g_1g_2)(a) = \sigma_{g_1g_2}(a) = (g_1g_2)\cdot a = g_1\cdot(g_2\cdot a) = (\sigma_{g_1} \circ \sigma_{g_2})(a) = (\varphi(g_1)\circ\varphi(g_2))(a) $$
Therefore, \(\varphi\) is a homomorphism.
$$\tag*{$\blacksquare$}$$