Distinct Powers of Elements in a Group

Distinct Powers of Elements in a Group

Recall from the The Order of Elements in a Group page that if $(G, *)$ is a group and $a \in G$ then the order of $a$ is the least positive integer $m$ such that $a^m = (\underbrace{a * a * … * a}_{m \: \mathrm{many \: factors}}) = e$ where $e$ is the identity element, and if no such positive integer $m$ exists then $a$ is said to have order $/infty$.

We will now look at a rather nice theorem which will tell us that if $a$ has order $m$, then the nonnegative powers of $a$ less than $m$ are all distinct elements in the group.

Theorem 1: Let $(G, *)$ be a group and let $a \in G$ have order $m$. Then the nonnegative integer powers less than $m$ are distinct, that is $a^0, a^1, …, a^{m-1}$ and these are the only powers of $a$.
  • Proof: Let $(G, *)$ be a group and let $a \in G$ have order $m$. Then $m$ is the least positive integer such that $a^m = e$ where $e$ is the identity element of $G$. We will begin by showing that these powers $a^0, a^1, …, a^{m-1}$ are the other powers of $a$. Let $s$ be any nonnegative integer. By The Division Algorithm we have that for integers $q$ and $r$ such that $0 \leq r < m$ that represent the quotient and remainder of $s$ upon division by $m$ that:
(1)
\begin{align} \quad s = mq + r \end{align}
  • Therefore:
(2)
\begin{align} \quad a^s = a^{mq + r} = a^{mq} * a^r = (a^m)^q * a^r = e^q * a^r = e * a^r = a^r \end{align}
  • Hence $a^s = a^r$, but $r \in \{ 0, 1, …, m - 1 \}$ which proves that $a^0, a^1, …, a^{m-1}$ are the only powers of $a$.
  • We will now show that the powers $a^0, a^1, …, a^{m-1}$ are distinct. Suppose that for some $s, t \in \{ 0, 1, …, m-1 \}$ we have that:
(3)
\begin{align} \quad a^s = a^t \end{align}
  • Without loss of generality, assume that $s \geq t$. Then:
(4)
\begin{align} \quad a^{s - t} = e \end{align}
  • Since $s, t < m$ we must have that $0 \leq s - t < m$. If $s - t \neq 0$ then this implies that the order of $a$ is $s - t < m$ which is a contradiction. Therefore $s - t = 0$ or rather, $s = t$. Hence $a^s = a^t$ if $s = t$, so the powers $a^0, a^1, …, a^{m-1}$ are distinct. $\blacksquare$
Theorem 2: Let $(G, *)$ be a group and suppose that $a \in G$ has order $\infty$. Then all powers of $a$ are distinct, that is $a^0, a^1, a^2, …$ are all distinct.
  • Proof: Let $(G, *)$ be a group and suppose that $a \in G$ has order $\infty$. Let $s$ and $t$ be nonnegative integers and suppose that:
(5)
\begin{align} \quad a^s = s^t \end{align}
  • Without loss of generality, assume that $s \geq t$. Then:
(6)
\begin{align} \quad a^{s - t} = e \end{align}
  • If $s - t > 0$ then this implies that $a$ has order $s - t$ which contradicts the hypothesis that $a$ has order $/infty$. Hence $s - t = 0$, so $s = t$. Therefore all of the powers $a^0, a^1, a^2, …$ of $a$ are unique. $\blacksquare$
Corollary 1: If $(G, *)$ is a finite group then for all $a \in G$, $a$ has a finite order.
  • Proof: Let $(G, *)$ be a group and suppose that for some $a \in G$ that $a$ has order $\infty$. Then all of the powers $a^0, a^1, a^2, …$ are distinct by Theorem 2. Since $(G, *)$ is closed under $*$ we must have that all of these powers are contained in $G$ contradicting the hypothesis that $(G, *)$ is a finite group. Hence every $a \in G$ has a finite order. $\blacksquare$
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