Maths - Cyclic Groups - Datasheet

For more information about cyclic groups see this page.

Cayley Table

The Cayley table is symmetric about its leading diagonal. For a cyclic group the table can be drawn with same terms on the bottom-left to top-right diagonals:

	1	a	b	c	...	n
1	1	a	b	c	...	n
a	a	b	c	...	n	n-1
b	b	c	...	n	n-1
c	c	...	n	n-1
...	...	n	n-1
n	n	n-1

For more information about Cayley table see this page.

Cayley Graph

For more information about Cayley graph see this page.

Cyclic Notation

The group is shown as a single cycle:

(1,2,3..n)

For more information about cyclic notation see this page.

Group Presentation

There is only one generator which when applied n times cycles back to the identity.

<a | aⁿ=1>

For more information about group presentation see this page.

Group Representation

This is the n^th root of the identity matix (such that lesser roots are not identity). See this page for information about taking roots of a matrix. One a matrix which will do this is an n×n matrix of this form:

0	0	…	0	0	1
1	0	…	0	0	0
0	1	…	0	0	0
0	0	…	1	0	0
0	0	…	0	1	0

For more information about group representation see this page.

Related datasheets

• Dihedral group.
• Complex Numbers

EuclideanSpace

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Group Theory