Permutation Diagrams and Wreath Products

This page shows how permutation diagrams can be applied to wreath products.

A definition of "wreath product"

A wreath product G≀H is formed from a group G and a permutation group H acting on a set of size N. We take the direct product of N copies of G, and form its semidirect product by H, where the action of H is to permute the N copies.

Example: a permutation diagram for C2≀C3


Figure 1a	Figure 1b	Figure 1c	Figure 1d

A permutation diagram for the wreath product C2≀C3, with the C3 performing cyclic permutations of three copies of C2, can be constructed as follows.

We start with figure 1a, a permutation diagram for C2×C2×C2. The generating elements of the three C2s are shown in pink, red, and orange.

Then we add the permuting C3, giving figure 1b. The black generator permutes the C2s in the order (pink, red, orange).

Next we observe that the three C2s are conjugate. We can therefore remove two of them, to give figure 1c.

Finally we rearrange the diagram, in the interests if symmetry and tidiness, to give figure 1d. This shows the group C2≀C3, which is is A4×C2, here acting on six points.

General method

We can use this as a general method for drawing a permutation diagram for a wreath product. The method, for the wreath product of G by (H acting on N points) is:

draw a permutation diagram for G (acting on any possible number of points)
draw a permutation diagram for H acting on N points, and sharing one point with that for G
copy the permutation diagram for H so that there is a copy of it attached to each point of the permutation diagram for G, and all the same way round.


Figure 2a	Figure 2b	Figure 2c	Figure 2d

This method has been applied in the last two figures to the right.
Figure 2a shows C3, acting on three points.
Figure 2b shows A4, acting on four points.
Figure 2c shows (C3 ≀ (A4 acting on four points)), with a total of 12 points.
Figure 2d shows ( (A4 acting on four points) ≀ C3), with a total of 12 points.

When we thus form the wreath product of (G acting on M points) by (H acting on N points), there will be a total of MN points.

Permutation diagrams of Sylow-2-subgroups of symmetric, and of alternating, groups.

The Sylow-2-subgroup of the symmetric group on 2ⁱ points is an iterated wreath product of C2 by C2. For i=1, this is C2; for i=2, it is D8; for i=3, it is D8 ≀ C2; for i=4, it is (D8 ≀ C2) ≀ C2; etc. The table below lists more such results.

n	Sylow-2-subgroup of S_n			Sylow-2-subgroup of A_n
n	size	permutation diagram	description	size	permutation diagram	description
0, 1	1		trivial group	1		trivial group
2, 3	2		C2	1		trivial group
4, 5	8		D8	4		C2×C2
6, 7	2⁴ = 16		D8 × C2	2⁴ = 8		D8
8, 9	2⁷ = 128		D8 ≀ C2	2⁶ = 64		(D8 ≀ C2) / C2
10, 11	2⁸ = 256		(D8 ≀ C2) × C2	2⁷ = 128		D8 ≀ C2
12, 13	2¹⁰ = 1,024		(D8 ≀ C2) × D8	2⁹ = 512		(D8, C2) ≀ C2
14, 15	2¹¹ = 2,048		(D8 ≀ C2) × D8 × C2	2¹⁰ = 1,024		(D8 ≀ C2) × D8
16, 17	2¹⁴ = 16,384		(D8 ≀ C2) ≀ C2	2¹³ = 8,192		((D8 ≀ C2) ≀ C2) / C2
18, 19	2¹⁵ = 32,768		((D8 ≀ C2) ≀ C2) × C2	2¹⁴ = 16,384		((D8 ≀ C2) ≀ C2)
20, 21	2¹⁷ = 131,072		((D8 ≀ C2) ≀ C2) × D8	2¹⁶ = 65,536		(((D8 ≀ C2) ≀ C2) × D8) / C2
22, 23	2¹⁸ = 262,144		((D8 ≀ C2) ≀ C2) × D8 × C2	2¹⁷ = 131,072		((D8 ≀ C2) ≀ C2) × D8
24, 25	2²¹ = 2,097,152		((D8 ≀ C2) ≀ C2) × (D8 ≀ C2)	2²⁰ = 1,048,576		(((D8 ≀ C2) ≀ C2) × (D8 ≀ C2)) / C2
26, 27	2²² = 4,194,304		((D8 ≀ C2) ≀ C2) × (D8 ≀ C2) × C2	2²¹ = 2,097,152		((D8 ≀ C2) ≀ C2) × (D8 ≀ C2)
28, 29	2²⁴ = 16,777,216		((D8 ≀ C2) ≀ C2) × (D8 ≀ C2) × D8	2²³ = 8,388,608		(((D8 ≀ C2) ≀ C2) × (D8 ≀ C2) × D8) / C2
30, 31	2²⁵ = 33,554,432		((D8 ≀ C2) ≀ C2) × (D8 ≀ C2) × D8 × C2	2²⁴ = 16,777,216		((D8 ≀ C2) ≀ C2) × (D8 ≀ C2) × D8
32, 33	2³⁰ = 1,073,741,912		((D8 ≀ C2) ≀ C2) ≀ C2	2²⁹ = 536,870,912		((D8 ≀ C2) ≀ C2) ≀ C2) / C2