Group

Groups are one of the most fundamental algebraic structures that underpin the study of abstract algebra. Simplistically, it is a set $$G$$ with an operation $$\circ$$ that satisfy the following axioms:


 * 1) Closure: For all $$g, h \in G$$, $$g \circ h \in G$$.
 * 2) Associativity: For all $$g, h, j \in G$$, $$ g \circ (h \circ j) = (g \circ h) \circ j$$.
 * 3) Identity: There exists an element $$e$$ such that, for all $$a \in G$$, $$a \circ e = a = e \circ a$$.
 * 4) Inverses: For all $$g \in G$$, there exists an element $$h \in G$$ such that $$g \circ h = e = h \circ g$$. We call this the inverse of $$g$$ and this inverse is unique.

Examples
To give some concreteness into the definition, we shall take a look at some examples of groups.


 * 1) Take the group of real numbers under addition. We write this as a tuple: $$(\mathbb R, +)$$. This is clearly a group since, for any two elements in $$\mathbb R$$, its sum is also in $$\mathbb R$$. It is also associative; the identity is simply the element 0 which is a real number; finally, every element $$a$$ has an inverse $$-a$$ which is also a real number. Hence, it is a group.
 * 2) Take the group of non-zero complex numbers under multiplication. We often write this using the following notation: $$\mathbb C^*$$ or $$\mathbb C^{\times}$$.
 * 3) The set of $$n \times n$$ invertible matrices with complex coefficients $$\operatorname{GL}_n(\mathbb C)$$ forms a group under standard matrix multiplication.

Properties
Let $$G$$ be a group endowed with the operation $$*$$. Then


 * 1) $$G$$ has a unique identity: that is, if $$ e, e'$$ are the identity elements of $$G$$, then $$e = e'$$.
 * Suppose that $$e, e'$$ are the identity elements of $$G$$. Then, for all $$g \in G$$, we have that $$g = e * g$$ and $$g = e' * g$$. This implies that $$e * g = e' * g$$. Multiplying both sides by the inverse of $$g$$, we obtain $$\left(e * g\right)g^{-1} = \left(e' * g\right)g^{-1} \implies e = e'$$.