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GroupTheory

 IsHomocyclic
 attempt to determine whether a group is homocyclic

 Calling Sequence IsHomocyclic( G )

Parameters

 G - a group

Description

 • A group $G$ is homocyclic if it is isomorphic to a direct power of a cyclic group; that is, of the form ${C}_{n}^{k}$, for some positive integer $n$ and non-negative integer $k$.
 • The IsHomocyclic( G ) command attempts to determine whether the group G is homocyclic.  It returns true if G is homocyclic and returns false otherwise. The command may return FAIL on (most) finitely presented groups.
 • Cyclic groups and elementary abelian groups are homocyclic.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsHomocyclic}\left(\mathrm{TrivialGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsHomocyclic}\left(\mathrm{CyclicGroup}\left(15\right)\right)$
 ${\mathrm{true}}$ (2)
 > $G≔\mathrm{SmallGroup}\left(36,14\right):$
 > $\mathrm{IsHomocyclic}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(6\right),\mathrm{CyclicGroup}\left(6\right)\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsHomocyclic}\left(\mathrm{SmallGroup}\left(4,1\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsHomocyclic}\left(⟨⟨a,b⟩|⟨{a}^{2},{b}^{2},a·b=b·a⟩⟩\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsHomocyclic}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsHomocyclic}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{false}}$ (8)

Compatibility

 • The GroupTheory[IsHomocyclic] command was introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.