
# 3.2: Subgroups

When we consider the symmetries of, say, a pentagon, we notice that it has rotational symmetries like the 'bumpy' pentagon. From the bumpy pentagon, we see that the rotations themselves form a group; there's a group of rotations inside the group of symmetries of the pentagon! Likewise, if we consider just the flip, we see a group similar to the symmetries of the perfectly symmetrical face. This yields another group inside the symmetries of the pentagon. We can make this precise:

Definition 3.2.0: Subgroup

Let $$G$$ be a group, and $$H$$ a subset of $$G$$. Then $$H$$ is a subgroup of $$G$$ if $$H$$ is itself a group using the same operation as $$G$$.

Ostensibly, to check that a subset $$H$$ is a subgroup, we would need to check all four properties of the group. That is, closure (ie, the operation gives a map $$H\times H\rightarrow H$$; products of things in $$H$$ are always in $$H$$), identity, the existence of inverses, and associativity.

In fact, since $$H$$ has the same operation as $$G$$, we know that the operation in $$H$$ is associative (since $$G$$ is a group). Furthermore, if the operation is closed and inverses exist, then we know that for any $$h\in H$$, $$hh^{-1}=e$$ must be in $$H$$. So really we only need to check two things:

1. Closure: $$gh\in H$$ for all $$g,h\in H$$, and
2. Inverses: $$h^{-1}\in H$$ for all $$h\in H$$.

Some important things to notice:

1. The group $$G$$ is always a subgroup of itself! ($$G$$ is a subset of itself, which is a group with the same operation as $$G$$.)
2. The subset containing just the identity element is also a subgroup! This is called the trivial subgroup.
3. The set of all powers of an element $$h$$ ($$\{\ldots, h^{-1}, h^{-2}, e, h, h^2, \ldots\}$$) is a subgroup of $$G$$. This is called the cyclic subgroup generated by $$h$$.

Exercise 3.2.1

Let $$X$$ be a geometric object. Show that the rotations of $$X$$ back onto itself forms a subgroup of the group of symmetries of $$X$$. (Try this in particular on a regular polygon and a regular polyhedron. What happens with a 'bumpy' polygon?)

Let $$G$$ be a group, and $$g\in G$$. Consider a function $$f_g:G\rightarrow G$$ given by $$f_g(h)=g\cdot h$$. (This is the 'left multiplication by $$g$$' function.) What happens if, for some $$h, k \in G$$, $$f_g(h)=f_g(k)$$? Then $$gh=gk$$, so $$g^{-1}gh=g^{-1}gk$$, and $$h=k$$. This tells us that $$f_g$$ is a one-to-one, or injective, function. If $$G$$ has a finite number of elements, then $$f_g$$ is also an onto function, and is thus a bijection from $$G$$ back to itself. Then we can consider $$f_g$$ as a permutation of $$G$$!

If we consider $$G$$ as a set, we can think of any left multiplication as a permutation of $$G$$. But the set of all left multiplications is itself a group. This gives us what is known as Cayley's Theorem!

Theorem 3.2.2: Cayley's Theorem

The ideal gas law is easy to remember and apply in solving problems, as long as you get the proper values a

Exercise 3.2.3

Label the six symmetries of the equilateral triangle. Demonstrate that the symmetries of the triangle are a subgroup of $$S_6$$, the permutations of $$6$$ objects.

It is worth noticing that for any $$g$$ in a group $$G$$, the powers of $$g$$ generate a subgroup of $$G$$. The set $$\{g^i \mid i \in \mathbb{Z} \}$$ is closed under the group operation, and includes the identity and inverses. This is called the cyclic subgroup generated by $$g$$.

Exercise 3,2,4

Find all of the subgroups of the permutation group $$S_3$$ for three objects. Which subgroups are subgroups of other subgroups? Name each subgroup, and arrange them according to which is contained in which.

### Contributors

• Tom Denton (Fields Institute/York University in Toronto)