DG7, DG9 - Groups and subgroups

Finite/infinite groups

If a square can be turned or reflected, do the turns alone still form a group? To answer, we need to describe both the full collection of moves and smaller collections within it.

A clock with twelve positions gives a finite example: adding positions modulo $12$ has the twelve elements $0,1,\ldots,11$. Ordinary integer addition gives an infinite example: there is no last integer to count. Both have identity $0$, but their sets have different sizes.

For $(\mathbb Z,+)$, sums are integers, the identity is $0$, every integer $a$ has inverse $-a$, and associativity is inherited from ordinary addition. The even integers also form an infinite group under addition: adding or negating an even integer stays even. Restricting the allowed elements has preserved all four axioms.

An infinite group need not give every element infinite period. In $(\mathbb R\setminus\{0\},\times)$, the element $-1$ has period $2$, because $(-1)^2=1$, although the group has infinitely many elements. Group size and element period remain different ideas.

Use that distinction in a new infinite group.

Polygon symmetries

A symmetry of a regular polygon moves it onto exactly the same outline. The elements are transformations, rather than the vertices themselves. Combining two symmetries means performing them in succession.

We use the convention $uv=u\circ v$: perform $v$ first, then $u$. This agrees with matrices acting on column vectors, where $\mathbf U\mathbf V\mathbf x$ applies $\mathbf V$ first. A question may specify another convention; follow its definition.

For a regular $n$-gon with $n\ge3$, there are $n$ rotations, including the identity, and $n$ reflections. We call its full symmetry group $D_n$, so $|D_n|=2n$. The meaning of the notation is fixed here because other texts sometimes use a different subscript. Composing symmetries is closed, doing nothing is the identity, reversing a movement gives its inverse, and function composition is associative. Thus these transformations form a group.

For the square, label $A=(1,1)$, $B=(-1,1)$, $C=(-1,-1)$, $D=(1,-1)$. Let $r$ be rotation anticlockwise through $90^\circ$, and let $s$ be reflection in the $y$-axis. Then

The eight symmetries can be listed as $e,r,r^2,r^3,s,rs,r^2s,r^3s$. Powers mean repeated composition.

[DIAGRAM: asset_name: Reversing two square symmetries; asset_slug: aqa7367-square-composition; caption: Reflection followed by rotation sends A to C, while rotation followed by reflection sends A to A; description: Two sequences of three identical coordinate squares show the labelled vertex A travelling under s then r in the top row, and r then s in the bottom row. The square has A at (1,1), B at (minus 1,1), C at (minus 1,minus 1), D at (1,minus 1). Blue dots track the original A: top row positions (1,1), (minus 1,1), (minus 1,minus 1); bottom row positions (1,1), (minus 1,1), (1,1). Headings state rs means s first then r, and sr means r first then s.]

Keep the same two moves and reverse only their order. The result can change even though each individual move is unchanged.

This gives a geometric group where commutativity fails, without any failure of the group axioms.

Try the same reasoning with a different polygon and explicitly supplied moves.

Cyclic and abelian groups

Repeated quarter-turns produce every rotational symmetry of a square: $e,r,r^2,r^3$, then $r^4=e$. Reflections never appear in this sequence. The rotations form a cyclic group, while the full square symmetry group does not.

In additive language, integer powers become integer multiples. Thus $(\mathbb Z,+)$ is infinite cyclic: all its elements are integer multiples of $1$. The group $(\{0,1,2,3\},+_4)$ is finite cyclic: repeating $1$ gives $1,2,3,0$.