Subgroups

Let \(G\) be a group and let \(H\) be a subset of \(G\). Then \(H\) is called a subgroup of \(G\) if \(H\) itself forms a group under the binary operation inherited from \(G\). This is written as

\[H\leq G.\]

The subgroup \(\{e\}\), consisting only of the identity element, is called the trivial subgroup of \(G\).

A subgroup \(H\) of \(G\) is called a proper subgroup if \(H\neq G\). It is commonly denoted by \(H<G\).

Subgroup Tests

Proposition: Let \(G\) be a group and let \(H\) be a nonempty subset of \(G\). Then \(H\) is a subgroup of \(G\) if and only if it satisfies either of the following equivalent conditions:

The second condition is commonly known as the one-step subgroup test.

Finite Subgroup Test

Proposition: Let \(G\) be a group and let \(H\) be a nonempty finite subset of \(G\). Then \(H\) is a subgroup of \(G\) if and only if

\[ab\in H\qquad\text{for all }a,b\in H.\]

Thus, for a nonempty finite subset, closure under the group operation is sufficient to establish that it is a subgroup.

Cyclic Subgroup Generated by an Element

Let \(a\in G\). The set of all integral powers of \(a\) is defined by

\[\langle a\rangle=\{a^k:k\in\mathbb{Z}\}.\]

This set forms a subgroup of \(G\) and is called the cyclic subgroup generated by \(a\). It is the smallest subgroup of \(G\) containing \(a\).

Kernel and Image as Subgroups

Proposition: Let \(\phi:G\rightarrow H\) be a group homomorphism. Then:

Center of a Group

The center of a group \(G\), denoted by \(Z(G)\), is the set of all elements of \(G\) that commute with every element of \(G\). Thus,

\[Z(G)=\{a\in G:ax=xa\text{ for every }x\in G\}.\]

Proposition: The center \(Z(G)\) is a subgroup of \(G\).

If \(G\) is abelian, then every element commutes with every other element, and hence

\[Z(G)=G.\]

Product of Two Subsets

For any two subsets \(A\) and \(B\) of a group \(G\), their product is defined by

\[AB=\{xy:x\in A,\ y\in B\}.\]

Proposition: Let \(H\) and \(K\) be subgroups of a group \(G\). Then \(HK\) is a subgroup of \(G\) if and only if

\[HK=KH.\]

Corollary: If \(H\) and \(K\) are subgroups of an abelian group \(G\), then \(HK\) is a subgroup of \(G\).

Right and Left Cosets

Let \(H\leq G\) and let \(a\in G\). The set

\[Ha=\{ha:h\in H\}\]

is called the right coset of \(H\) in \(G\) determined by \(a\).

Similarly, the set

\[aH=\{ah:h\in H\}\]

is called the left coset of \(H\) in \(G\) determined by \(a\).

Every left or right coset of \(H\) has the same number of elements as \(H\). In particular, there is a one-to-one correspondence between \(H\) and each of its cosets.

Basic Properties of Cosets

Let \(H\leq G\) and let \(a,b\in G\). Then:

Exercise: Show that there is a one-to-one correspondence between the set of left cosets of \(H\) in \(G\) and the set of right cosets of \(H\) in \(G\).

Lagrange’s Theorem

Theorem: Let \(G\) be a finite group and let \(H\leq G\). Then the order of \(H\) divides the order of \(G\). In symbols,

\[|H|\mid |G|.\]

More precisely,

\[|G|=[G:H]\,|H|,\]

where \([G:H]\) is the number of distinct left cosets, or equivalently the number of distinct right cosets, of \(H\) in \(G\).

Index of a Subgroup

Let \(H\leq G\). The index of \(H\) in \(G\) is the number of distinct right cosets of \(H\) in \(G\). It is denoted by

\[[G:H].\]

The number of distinct right cosets is equal to the number of distinct left cosets. Therefore, either collection may be used to define the index.

Order of an Element

Let \(G\) be a group with identity element \(e\), and let \(a\in G\). If there exists a least positive integer \(m\) such that

\[a^m=e,\]

then \(m\) is called the order of \(a\), denoted by \(o(a)\) or \(|a|\).

If no such positive integer exists, then \(a\) is said to have infinite order.

The order of \(a\) is equal to the order of the cyclic subgroup generated by \(a\). Thus,

\[o(a)=|\langle a\rangle|.\]

Consequences of Lagrange’s Theorem