Groups of order p³

For every prime p, there are five non-isomorphic groups of order p³. Three are Abelian, these are Cp³, Cp²×Cp, and Cp×Cp×Cp. Two are non-Abelian. The rest of this page is about the non-Abelian groups.

Non-Abelian Groups of order p³

For any prime p, we can imagine three non-Abelian groups of order p:

• (Cp×Cp) ⋊ Cp
• (Cp²) ⋊ Cp
• Cp ↑ (Cp×Cp)

However, for all values of p, there turn out to be only two. Which two depends on whether p is even or odd.

Non-Abelian Groups of order p³, p an even prime; i.e. of order 8

For p=2, the two groups (C2×C2)⋊C2 and C4⋊C2 are isomorphic. This is the group usually known as D8 (or by some authors as D4). It can be presented as either
    < a, b | a⁴, b², abab >
or as
    < a, b, t | a², b², t², atb³t, bta³t >
It has five elements of order 2, falling into two equivalence classes of sizes one and four, and two elements of order four.

However C2 ↑ (C2×C2) is another group of order eight, with no analogue for higher values of p. This is the quaternion group. It can be presented as
    < i, j | i⁴, j⁴, iji³j, i²j² >
It has one elements of order 2, and six elements of order four forming a single equivalence class.

Non-Abelian Groups of order p³, p an odd prime

For every odd prime p, (Cp×Cp) ⋊ Cp is a group which can be presented as
    < a, b | a^p2, b^p, bab^p-1a^p2-p-1 >
It has p²-1 elements of order p, which fall into two equivalence classes, of sizes p-1 and p²-p; and p³-p² elements of order p², forming a single equivalence class.

For every odd prime p, there is also a group (Cp²) ⋊ Cp. It has p³-1 elements of order p, which fall into three equivalence classes, of sizes p-1, p²-p, and p³-p². Cp ↑ (Cp×Cp) can be interpreted in various ways, but each of them is isomorphic to one of (Cp×Cp) ⋊ Cp, (Cp²) ⋊ Cp, or Cp² × Cp.

Summary

Non-Abelian groups of order p³.

More miscellaneous short pages on finite groups
More pages on groups

Copyright N.S.Wedd 2008, 2009