A subset \(H\) of a group \(G\) is called a subgroup of \(G\) if \(H\) itself is a group under the group operation of \(G\) restricted to \(H\text{.}\) We write \(H\leq
G\) to indicate that \(H\) is a subgroup of \(G\text{.}\) Given a subgroup \(H\) of \(G\text{,}\) and given an element \(g\) in \(G\text{,}\) the set

Let \(H\) be a subset of a group \(G\text{.}\) The following are equivalent.

\(H\) is a subgroup of \(G\)

(2-step subgroup test) \(H\) is nonempty, \(ab\) is in \(H\) for every \(a,b\) in \(H\) (\(H\) is closed under the group operation), and \(a^{-1}\) is in \(H\) for every \(a\) in \(H\) (\(H\) is closed under group inversion)

(1-step subgroup test) \(H\) is nonempty and \(ab^{-1}\) is in \(H\) for every \(a,b\) in \(H\)

Proposition2.3.5.Subgroup generated by a set of elements.

Let \(S\) be a nonempty subset of a group \(G\text{,}\) and let \(S^{-1}\) denote the set \(S^{-1}=\{s^{-1}\colon s\in S\}\) of inverses of elements in \(S\text{.}\) We write \(\langle S\rangle\) to denote the set of all elements of \(G\) of the form

where \(k\) ranges over all positive integers and each \(s_i\) is in \(S\cup S^{-1}\) for \(1\leq i\leq k\text{.}\) The set \(\langle S\rangle\) is a subgroup of \(G\text{,}\) called the subgroup generated by the set \(S\) , and the elements of \(S\) are called the generators of \(\langle S\rangle\text{.}\)

Comment on notational convention: If \(S=\{s_1,s_2,\ldots,s_k\}\) is finite, we write \(\langle
s_1,s_2,\ldots,s_k\rangle\) for \(\langle S\rangle\text{,}\) instead of the more cumbersome \(\langle \{s_1,s_2,\ldots,s_k\}\rangle\text{.}\)

Observation2.3.6.

If \(G\) is a cyclic group with generator \(g\text{,}\) then \(G=\langle g\rangle\text{.}\)

Checkpoint2.3.7.

Show that \(\langle S\rangle\) is indeed a subgroup of \(G\text{.}\) How would this fail if \(S\) were empty?

Let \(G\) be a group and let \(H\) be a subgroup of \(G\text{.}\) Let \(\sim_H\) be the relation on \(G\) defined by \(x\sim_H y\) if and only if \(x^{-1}y \in
H\text{.}\) The relation \(\sim_H\) is an equivalence relation on \(G\text{,}\) and the equivalence classes are the cosets of \(H\text{,}\) that is, we have \(G/\!\!\sim_H= G/H\text{.}\)

Corollary2.3.10.Cosets as a partition.

Let \(G\) be a group and let \(H\) be a subgroup of \(G\text{.}\) The set \(G/H\) of cosets of \(H\) form a partition of \(G\text{.}\)

In the "list of values" permutation notation of Checkpoint 2.1.2, the subgroups of \(S_3\) are \(\{[1,2,3]\}\text{,}\)\(\{[1,2,3],[2,1,3]\}\text{,}\)\(\{[1,2,3],[1,3,2]\}\text{,}\)\(\{[1,2,3],[3,2,1]\}\text{,}\)\(\{[1,2,3],[2,3,1],[3,1,2]\}\text{,}\) and \(S_3\text{.}\) In cycle notation, the subgroups of \(S_3\) (in the same order) are \(\{e\}\text{,}\)\(\{e,(12)\}\text{,}\)\(\{e,(23)\}\text{,}\)\(\{e,(13)\}\text{,}\)\(\{e,(123),(132)\}\text{,}\)\(S_3\text{.}\)

3.

Find all the cosets of the subgroup \(\{R_0,R_{1/2}\}\) of \(D_4\text{.}\)

4.Subgroups of \(\Z\) and \(\Z_n\).

Let \(H\) be a subgroup of \(\Z\text{.}\) Show that either \(H=\{0\}\) or \(H=\langle d\rangle\text{,}\) where \(d\) is the smallest positive element in \(H\text{.}\)

Let \(H\) be a subgroup of \(\Z_n\text{.}\) Show that either \(H=\{0\}\) or \(H=\langle d\rangle\text{,}\) where \(d\) is the smallest positive element in \(H\text{.}\)

Let \(n_1,n_2,\ldots,n_r\) be positive integers. Show that

Consequence of this exercise: The greatest common divisor \(\gcd(a,b)\) of integers \(a,b\) is the smallest positive integer of the form \(sa+tb\) over all integers \(s,t\text{.}\) Two integers \(a,b\) are relatively prime if and only if there exist integers \(s,t\) such that \(sa+tb=1\text{.}\)

5.Centralizers, Center of a group.

The centralizer of an element \(a\) in a group \(G\text{,}\) denoted \(C(a)\text{,}\) is the set

Show that the centralizer \(C(a)\) of any element \(a\) in a group \(G\) is a subgroup of \(G\text{.}\)

Show that the center \(Z(G)\) of a group \(G\) is a subgroup of \(G\text{.}\)

6.The order of a group element.

Let \(g\) be an element of a group \(G\text{.}\) The order of \(g\text{,}\) denoted \(|g|\text{,}\) is the smallest positive integer \(n\) such that \(g^n=e\text{,}\) if such an integer exists. If there is no positive integer \(n\) such that \(g^n=e\text{,}\) then \(g\) is said to have infinite order. Show that, if the order of \(g\) is finite, say \(|g|=n\text{,}\) then

\begin{equation*}
\langle g \rangle =
\{g^0,g^1,g^2,\ldots,g^{n-1}\}\text{.}
\end{equation*}

Consequence of this exercise: If \(G\) is cyclic with generator \(g\text{,}\) then \(|G|=|g|\text{.}\)

7.Cosets of a subgroup partition the group, Lagrange’s Theorem.