Common Definition

Let \(G\) be a group - infinite or finite, either is fine. Let \(x\in G\) be any element. If you are using multiplicative notation, then we say the order of \(x\) is the smallest positive integer \(n\)where \(x^n=1\). \[\\\] If you are using additive notation, then we say the order of \(x\) is the smallest positive integer \(n\) where \(n\cdot x = 0\). \[\\\] In either case we write \(|x|=n\) and say this aloud as "the order of \(x\) is \(n\)".

Alternative Definition

Let \(G\) be ANY group and \(x\in G\). Look at the sequence of powers: \(x, x^2, x^3, ...\). There are two possible outcomes: it's either an infinite sequence of distinct elements, or eventually the sequence repeats. \[ \\ \] If the sequence repeats with a cycle of length \(n\), we say \(x\) is of order \(n\) and we write this as \(|x|=n\). \[ \\ \] If the sequence never repeats we say \(x\) has \(\textit{infinite order}\).

Examples

  • The number \(i\)

    The number \(i=\sqrt{-1}\) has order 4 because \(i^4=1\) and this is the smallest positive integer for which this is true. \(i^8=1, i^{12}=1, ...\) but these are not the smallest positive integers that work.

  • \(2 \pmod{10}\)

    The integers mod 10 form a group under addition. The order of \(2\) is \(5\) because \[5\cdot 2=0 \pmod{10}\]

  • A Matrix

    \[ A = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix} \] This matrix has order 3: \(|A|=I\) where \(I\) is the identity matrix.

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Related Lesson

Kernel of Group Homomorphisms

In abstract algebra, the concept of a homomorphism is used to compare two structures—such as groups, rings, or fields. Homomorphisms are often not one-to-one, and one way to find how far a homomorphism is from being injective is by finding its kernel.
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