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:

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, bta3>
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 | ap2, bp, babp-1ap2-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.

pCp2 ⋊ Cp(Cp×Cp) ⋊ CpCp ↑ (Cp×Cp)
2C4⋊C2  ≅  D8  ≅  (C2×C2)⋊C2Q8
3C9⋊C3(C3×C3)⋊C3
5C25⋊C5(C5×C5)⋊C5
7C49⋊C7(C7×C7)⋊C7
11C121⋊C11(C11×C11)⋊C11
 etc.etc.

More miscellaneous short pages on finite groups
More pages on groups

Copyright N.S.Wedd 2008, 2009

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