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GroupTheory

 Exponent
 compute the exponent of a finite group

 Calling Sequence Exponent(G)

Parameters

 G - a permutation group or a Cayley table group

Description

 • The exponent of a finite group G is the least positive integer e such that g^e = 1, for all g in G. It is equal to the least common multiple of the orders of the elements of G, and is a divisor of the order of G.
 • The Exponent(G) command computes the exponent of the finite group G, if possible.
 • Note that the exponent of a finite finitely presented group can be computed by first converting it to a permutation group using the PermutationGroup command.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SymmetricGroup}\left(5\right)$
 ${G}{≔}{{\mathbf{S}}}_{{5}}$ (1)
 > $\mathrm{Exponent}\left(G\right)$
 ${60}$ (2)
 > $\mathrm{Exponent}\left(\mathrm{ElementaryGroup}\left(7,3\right)\right)$
 ${7}$ (3)
 > $G≔⟨⟨a,b⟩|⟨{a}^{2},{b}^{3},{\left(a·b\right)}^{5}=1⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{3}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}⟩$ (4)
 > $\mathrm{Exponent}\left(G\right)$
 > $\mathrm{Exponent}\left(\mathrm{PermutationGroup}\left(G\right)\right)$
 ${30}$ (5)

Compatibility

 • The GroupTheory[Exponent] command was introduced in Maple 2015.
 • For more information on Maple 2015 changes, see Updates in Maple 2015.