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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SymmetricGroup}\left(5\right)$

$G≔{\mathbf{S}}_5$

$\mathrm{Exponent}\left(G\right)$

$60$

$\mathrm{Exponent}\left(\mathrm{ElementaryGroup}\left(7\,3\right)\right)$

$7$

$\mathrm{Exponent}\left(\mathrm{QuasicyclicGroup}\left(2\right)\right)$

$\mathrm{\infty }$

$G≔⟨⟨a\,b⟩|⟨a^2\,b^3\,{\left(a·b\right)}^5=1⟩⟩$

$G≔⟨a\,b\mid a^2\,b^3\,ababababab⟩$

$\mathrm{Exponent}\left(G\right)$

Error, (in Exponent) cannot compute the Exponent of a general finitely presented group.  If you know that your group is finite, try converting it to a permutation group by using the `PermutationGroup' command with your finitely presented group as input.

$\mathrm{Exponent}\left(\mathrm{PermutationGroup}\left(G\right)\right)$

$30$

The GroupTheory[Exponent] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.