Cyclic group:Z10

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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This group is defined in the following equivalent ways:

  1. It is the cyclic group of order 10.
  2. It is the direct product of the cyclic group of order five and the cyclic group of order two.

Arithmetic functions

Function Value Explanation
order 10
exponent 10

GAP implementation

Group ID

This finite group has order 10 and has ID 2 among the groups of order 10 in GAP's SmallGroup library. For context, there are groups of order 10. It can thus be defined using GAP's SmallGroup function as:


For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(10,2);

Conversely, to check whether a given group G is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [10,2]

or just do:


to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

The group can be defined using GAP's CyclicGroup function: