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Group Theory

One of the first bits of ``math'' you learned as a student is ordinary arithmetic: how to add and subtract two numbers, how to multiply and divide two numbers. Although you may not have realized it at the time, you were learning not only your first arithmetic, but your first group theory! However, group theory is a lot more general than ``just'' arithmetic.

A group $ {\cal G}$ is a set of elements that is closed with respect to an operation of composition (think ``multiplication'', although it often isn't, so people use a multiplication-like symbol, $ \circ$ , instead of $ *$ when discussing abstract groups) that turns two elements into one (not necessarily different) element:

$\displaystyle a \circ b = c, \quad\quad \textrm{with } a,b,c\in {\cal G}$ (5.1)

The set of elements has to contain one special element, the identity element $ i$ , such that:

$\displaystyle a\circ i = a $

Every element must have a corresponding inverse element in the group:

$\displaystyle a \circ a^{-1} = i, \quad\quad \textrm{with } a,a^{-1},i\in {\cal G} $

Finally, the composition rule has to be associative:

$\displaystyle a\circ(b\circ c) = (a\circ b)\circ c, \quad\quad \textrm{with }
a,b,c\in {\cal G} $

The simplest, and smallest, group consists of only one element, the identity element, which is its own inverse, represented by a single line:

$\displaystyle i\circ(i\circ i) = (i\circ i)\circ i = i \circ i = i $

where we see that the identity element is always its own inverse and forms all by itself a special group called the trivial group. The trivial group is denoted $ Z_1$ (or sometimes $ C_1$ ).

You are familiar with a number of groups already, even though you may not have thought of them as such. The set of positive and negative integers, with the addition symbol used for composition, forms a group, with zero being the identity and a negative number being the inverse of a positive one and vice versa. The set of integers together with multiplication used as a composition rule is not a group! It is associate, it is closed, and it has an identity (the integer one) but the inverse of almost all elements is not in the group. The set of all rational numbers excluding zero forms a group with respect to multiplication (why must we exclude zero?). Mathematicians notationally write this exclusion with the $ \backslash$ symbol, for example the general multiplicative group over the set (field) of all complex numbers $ \mathbb{C}$ is denoted $ \mathbb{C}^* =
\mathbb{C}\backslash 0$ .



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Robert G. Brown 2017-07-11