Groups and subgroups · Definition

Generated subgroup

modern algebragroups

Before you start you need to know

It lets us build subgroups from a few elements and describe whole groups through their generators.

Definition

The subgroup generated by a subset \(S\) of a group \(G\) is the smallest subgroup \(H\) of \(G\) containing \(S\); it is denoted \(\langle S\rangle\) and \(S\) is called a generating set.

\[\langle S\rangle=\bigcap_{S\subseteq H\le G}H\]

The formula says that \(\langle S\rangle\) is the intersection of all subgroups containing \(S\); that intersection is a subgroup and is the smallest possible. From the inside, \(\langle S\rangle\) consists of all finite products of elements of \(S\) and their inverses, including the empty product, which is the identity. Hence \(\langle\varnothing\rangle=\{e\}\).

When \(S=\{a\}\), the subgroup is \(\langle a\rangle=\{a^k : k\in\mathbb{Z}\}\) and its number of elements is the order of \(a\). In \((\mathbb{Z},+)\), \(\langle m,n\rangle=\gcd(m,n)\,\mathbb{Z}\); in \(\mathbb{Z}_n\), \(\langle a\rangle\) has \(n/\gcd(a,n)\) elements. In \(S_3\), two distinct transpositions generate the whole group.

A group is called finitely generated if some finite \(S\) satisfies \(\langle S\rangle=G\). The list of generators is neither unique nor necessarily minimal: \(\mathbb{Z}\) is generated by \(\{1\}\) or by \(\{2,3\}\). When a single element generates the whole group, it is cyclic, the simplest kind of group.

Think about it

Why is the union of two subgroups not good for defining \(\langle S\rangle\)?

Because a union of subgroups is almost never closed under the operation; that is why one takes all products or, as in the formula, intersects subgroups.

In \((\mathbb{Z},+)\), which subgroup do \(\{2,3\}\) generate?

All of \(\mathbb{Z}\): since \(\gcd(2,3)=1\), we get \(1=3-2\) and from it every integer.

Exercises 10

  1. In the group \((\mathbb{Z},+)\), what is the subgroup generated by \(4\)?
  2. How many elements does \(\langle 8\rangle\) have in \((\mathbb{Z}_{12},+)\)?
  3. How many elements does \(\langle 4,6\rangle\) have in \((\mathbb{Z}_{12},+)\)?
  4. In \((\mathbb{Z},+)\), what is the subgroup \(\langle 6,15\rangle\)?
  5. How many elements does the subgroup of \(S_3\) generated by the 3-cycle \((1\ 2\ 3)\) have?
  6. How many elements does \(\langle (1\ 2),(2\ 3)\rangle\) have in \(S_3\)?
  7. How many elements does \(\langle (1\ 2),(3\ 4)\rangle\) have in \(S_4\)?
  8. What is the subgroup \(\langle\varnothing\rangle\) generated by the empty set in a group \(G\)?
  9. In \((\mathbb{Z}_{18},+)\), which of the following elements generates the same subgroup as \(4\)?
  10. Let \(a\) be an element of order 10 in a group. How many elements does \(\langle a^4\rangle\) have?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §2.4
  • Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 3 y 4
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §5