Groups

The notion of group is presumed.
The smallest group is one consisting of one element only, which, by definition of group, must be the identity element of the group. Any such group is called “trivial”. All such groups are isomorphic, but not identical. Here are several isomorphic, but not identical, such groups: \[\begin{array}{ll} &\text{underlying set} &\text{group operation}\\[1ex] &\{0 \in \Z\} &\text{addition}\\ &\{1 \in \Z\} &\text{multiplication}\\ &\{ 1_X : X \to X \} &\text{function composition, for any set } X\\ \end{array}\] Like the groups with one element, all groups with two elements are isomorphic.
If we denote the identity element by $e$ and the non-identity element by $t$,
such a group must have the multiplication table \[\begin{array}{ccc} &&e&t\\[1ex] e&&e&t\\ t&&t&e\\ \end{array}\] So the non-identity element is of order 2.

(Page down to see the remainder of this post.)

However, as with groups of order 1, the isomorphic groups of order 2 may have different underlying sets, and different group operations.
Here are some significant, and frequently encountered, examples: \[\begin{array}{cccccc} \text{name} & \text{underlying set} & \text{identity} & \text{non-identity} & \text{group operation} & \text{comment}\\[2ex] {\bf Z}_2 & \{0,1\} & 0 & 1 & \text{addition modulo 2} & \text{the integers modulo 2}\\ {\bf O}_1 & \{+1,-1\} & +1 & -1 & \text{multiplication} & \text{the one-dimensional orthogonal group}\\ S_2 & \text{bijections of }\{1,2\} \text{ with itself}& \text{identity function} & (1,2) & \text{function composition} & \text{symmetric group on two elements}\\ \end{array}\]

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To illustrate our notation (the cycle notation for permutations),
here are the elements of the symmetric group, or group of permutations, $S_3 = S_{[3]}$
of symmetries, or permutations, or autobijections, of $[3] \equiv \{0,1,2\}$.
Its $3!=6$ elements are denoted as follows:

• $\iota \equiv e \equiv (0)(1)(2)$, the identity function on $[3]$.
  When this function is considered in the context of all function,
  the usual notation for it is $1_{[3]}$.
  But when the context is only the group $S_3$,
  $\iota$ or $e$ are almost invariably used.
  
  Then the three 2-cycles:
  
• $(0,1)(2)$.
  It is common to omit the fixed points when writing permutations in cycle notation,
  so this becomes just $(0,1)$.
• $(0,2)(1) \equiv (0,2)$
• $(1,2)(0) \equiv (1,2)$.
  
  Then the two 3-cycles:
  
• $(0,1,2)$
• $(2,1,0)$