Skip to main content

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 \(G\text{,}\) the cyclic subgroup generated by \(g\) is the collection

\begin{equation*} \gen{g} = \{ \dotsc, \inv[3]{g}, \inv[2]{g}, \inv{g}, e, g, g^2, g^3, \dotsc \} \end{equation*}

consisting of all powers (positive, negative, and zero) of \(g\text{.}\) (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

\begin{equation*} \gen{g} = \{ \dotsc, -3 \cdot g, -2 \cdot g, -g, 0, g, 2 \cdot g, 3 \cdot g, \dotsc \} \text{.} \end{equation*}

Now let's extend this idea to a generating set that contains more than one element from the group \(G\text{.}\)

Definition 5.1.4. Subgroup generated by set \(A\).

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

\begin{equation*} a_1^{m_1} a_2^{m_2} \dotsm a_k^{m_k} \end{equation*}

for some collection of elements \(a_1,a_2,\dotsc,a_m\) from \(A\) and corresponding collection \(m_1,m_2,\dotsc,m_k\) of integer exponents.

We write \(\gen{A}\) to denote the subgroup of \(G\) generated by \(A\text{.}\) Elements of the set \(A\) are called generators of the subgroup \(\gen{A}\text{.}\)

Remark 5.1.5.

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

\begin{equation*} a_1, a_2, \dotsc, a_n \text{,} \end{equation*}

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

\begin{equation*} \gen{ a_1, a_2, \dotsc, a_n } \end{equation*}

instead of

\begin{equation*} \gen{ \{ a_1, a_2, \dotsc, a_n \} } \end{equation*}

for the subgroup generated by this collection of elements.

In additive notation, we would write that elements of \(\gen{A}\) take the form

\begin{equation*} m_1 \cdot a_1 + m_2 \cdot a_2 + \dotsb + m_k \cdot a_k \end{equation*}

for some collection of elements \(a_1,a_2,\dotsc,a_m\) from \(A\) and corresponding collection \(m_1,m_2,\dotsc,m_k\) of integer coefficients. In this context, it is very much like \(\gen{A}\) is actually \(\Span A\text{,}\) the collection of all possible “linear combinations” of elements from \(A\text{,}\) 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 \(\gen{g}\) and \(\gen{A}\) 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 \(\gen{A}\) in Definition 5.1.4.

  • It may be possible to realize a particular element in \(\gen{A}\) as a product of powers of elements from \(A\) in many different ways. Remember: what the collection \(\gen{A}\) really contains are the results of calculating out formulas involving products of powers of elements from \(A\text{,}\) 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 \(G\text{,}\) say

    \begin{equation*} A = \{ a_1, a_2, \dotsc, a_n \} \text{,} \end{equation*}
    then in general it is not the case that every element of \(\gen{A}\) will be of the form
    \begin{equation*} a_1^{m_1} a_2^{m_2} \dotsm a_n^{m_n} \text{,} \end{equation*}
    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 \(\gen{A}\text{.}\) For example, if \(A\) consists of two elements, say

    \begin{equation*} A = \{ a_1, a_2 \} \text{,} \end{equation*}
    then \(\gen{A}\) contains each of the results of calculating
    \begin{equation*} a_1 a_2 a_1, \qquad a_1^2 a_2, \qquad a_2 a_1^2, \end{equation*}
    and these results might all be different.