UA-AUG-AUMAT229 Pre-read

Section 5.1 Pre-read

Remark 5.1.1.

From now on we will mostly stick to multiplicative notation when discussing abstract concepts, but will comment on additive versions of things as appropriate.

🔗

Definition 5.1.2. Subgroup of a group.

A collection of elements from a group that, on its own, forms a group under the same binary operation as the “parent” group.

🔗

Just as we used spanning sets to create subspaces of vector spaces in linear algebra, we can take a collection of elements in a group that is not (necessarily) a subgroup and build a larger collection that contains the original collection and is a subgroup.

🔗

Let's start with a single element.

🔗

Definition 5.1.3. Cyclic subgroup generated by $g$ .

For element $g$ in group $G,$ the cyclic subgroup generated by $g$ is the collection

⟨ g ⟩ = {\dots, g^{- 3}, g^{- 2}, g^{- 1}, e, g, g^{2}, g^{3}, \dots}

consisting of all powers (positive, negative, and zero) of $g .$ (Recall that $g^{0}$ is taken to mean $e$ by convention.) The element $g$ is called the generator of the cyclic subgroup.

🔗

In additive notation, we would have

⟨ g ⟩ = {\dots, - 3 \cdot g, - 2 \cdot g, - g, 0, g, 2 \cdot g, 3 \cdot g, \dots} .

🔗

Now let's extend this idea to a generating set that contains more than one element from the group

G .

🔗

Definition 5.1.4. Subgroup generated by set $A$ .

For collection $A$ of elements in group $G,$ the subgroup generated by $A$ is the collection of all results of computing products of powers (positive, negative, and zero) of elements from $A .$ That is, each element of the subgroup generated by $A$ can be expressed as

a_{1}^{m_{1}} a_{2}^{m_{2}} \dots a_{k}^{m_{k}}

for some collection of elements $a_{1}, a_{2}, \dots, a_{m}$ from $A$ and corresponding collection $m_{1}, m_{2}, \dots, m_{k}$ of integer exponents.

We write $⟨ A ⟩$ to denote the subgroup of $G$ generated by $A .$ Elements of the set $A$ are called generators of the subgroup $⟨ A ⟩ .$

🔗

Remark 5.1.5.

If the generating collection contains a finite number of elements, say

a_{1}, a_{2}, \dots, a_{n},

and we haven't bothered to give this collection a name (like $A$ ), one usually writes simply

⟨ a_{1}, a_{2}, \dots, a_{n} ⟩

instead of

⟨ {a_{1}, a_{2}, \dots, a_{n}} ⟩

for the subgroup generated by this collection of elements.

🔗

In additive notation, we would write that elements of

⟨ A ⟩

take the form

m_{1} \cdot a_{1} + m_{2} \cdot a_{2} + \dots + m_{k} \cdot a_{k}

🔗

for some collection of elements

a_{1}, a_{2}, \dots, a_{m}

from

A

and corresponding collection

m_{1}, m_{2}, \dots, m_{k}

of integer coefficients. In this context, it is very much like

⟨ A ⟩

is actually

Span A,

the collection of all possible “linear combinations” of elements from

A,

except that we are limited to using integer scalars instead of real-valued scalars.

🔗

Remark 5.1.6.

Don't worry if you are not sure why the collections $⟨ g ⟩$ and $⟨ A ⟩$ are guaranteed to be subgroups — we will explore that in the discovery guide.

🔗

Warning 5.1.7.

Be careful about how you interpret the specification of the form of elements in $⟨ A ⟩$ in Definition 5.1.4.

It may be possible to realize a particular element in $⟨ A ⟩$ as a product of powers of elements from $A$ in many different ways. Remember: what the collection $⟨ A ⟩$ really contains are the results of calculating out formulas involving products of powers of elements from $A,$ and two such formulas might yield the same result.
Order of multiplication matters in a non-Abelian group! So if $A$ is a finite collection of elements from $G,$ say
$A = {a_{1}, a_{2}, \dots, a_{n}},$
then in general it is not the case that every element of $⟨ A ⟩$ will be of the form
$a_{1}^{m_{1}} a_{2}^{m_{2}} \dots a_{n}^{m_{n}},$
because that is imposing the same order of multiplication on all such formulas.
Again, since order of multiplication matters in a non-Abelian group, an element of $A$ may appear twice in a product-of-powers formula realizing some particular element of $⟨ A ⟩ .$ For example, if $A$ consists of two elements, say
$A = {a_{1}, a_{2}},$
then $⟨ A ⟩$ contains each of the results of calculating
$a_{1} a_{2} a_{1}, a_{1}^{2} a_{2}, a_{2} a_{1}^{2},$
and these results might all be different.