Why do this problem?
Groups play a key role in understanding the underlying patterns in
geometric art as well as underpinning arithmetic, algebra, geometry
and being useful in many applications of mathematics.
This challenge is structured to enable learners to find out for
themselves about the group of symmetries of a regular pentagon and
to find all of its subgroups.They don't have to have any prior
knowledge of groups and they do not need to know the axioms of a
group because with transformations there is always an inverse. They
only need to know that they are looking for sets of transformations
which are closed under the operation of taking combinations.
The class could start by listing the symmetries in the four
diagrams and discussing how to use symbols for the transformations.
Class discussion can then explore the key ideas:
(1) a combination of two reflections in intersecting lines gives a
rotation and so once a group contains two reflections it must
contain all the symmetries of the pentagon
(2) a group that contains a reflection and a rotation contains all
the symmetries of the pentagon
You can introduce the words 'subgroup' and 'isomorphic groups' as
the standard terminology for what the learners have discovered for
What happens when you combine two reflections in intersecting
What transformation results when you combine a reflection in a
mirror line and a rotation about a point on the mirror line?
What is the same and what is different about the groups with two
elements that contain the identity and reflection in a single
The challenge is so structured that it could be a short activity
leading on to a similar investigation of the symmetries of a
cylinder and finding out why there are only seven frieze groups.
See Paint Rollers for Frieze
To explore the ideas of groups of transformations further you could
read the article Grouping Transformations.
The three challenges: Decoding Transformations, Combining Transformations
and Simplifying Transformations
introduce the idea of an 'algebra' to record transformations and
the way transformations combine to give other transformations.