Cycle graph (algebra)
In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
A cycle is the set of powers of a given group element a, where an, the n-th power of an element a is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, e; the lowest such power is the order of the cycle, the number of distinct elements in it. In a cycle graph, the cycle is represented as a polygon, with the vertices representing the group elements, and the connecting lines indicating that all elements in that polygon are members of the same cycle.
Cycles
Cycles can overlap, or they can have no element in common but the identity. The cycle graph displays each interesting cycle as a polygon.
If a generates a cycle of order 6 (or, more shortly, has order 6), then a6 = e. Then the set of powers of a2, {a2, a4, e} is a cycle, but this is really no new information. Similarly, a5 generates the same cycle as a itself.
So, only the primitive cycles need be considered, namely those that are not subsets of another cycle. Each of these is generated by some primitive element, a. Take one point for each element of the original group. For each primitive element, connect e to a, a to a2, ..., an−1 to an, etc., until e is reached. The result is the cycle graph.
When a2 = e, a has order 2 (is an involution), and is connected to e by two edges. Except when the intent is to emphasize the two edges of the cycle, it is typically drawn[1] as a single line between the two elements.
Properties
 Dih4 kaleidoscope with red mirror and 4-fold rotational generators |
 Cycle graph for dihedral group Dih4. |
As an example of a group cycle graph, consider the dihedral group Dih4. The multiplication table for this group is shown on the left, and the cycle graph is shown on the right with e specifying the identity element.
| o | e | b | a | a2 | a3 | ab | a2b | a3b |
|---|---|---|---|---|---|---|---|---|
| e | e | b | a | a2 | a3 | ab | a2b | a3b |
| b | b | e | a3b | a2b | ab | a3 | a2 | a |
| a | a | ab | a2 | a3 | e | a2b | a3b | b |
| a2 | a2 | a2b | a3 | e | a | a3b | b | ab |
| a3 | a3 | a3b | e | a | a2 | b | ab | a2b |
| ab | ab | a | b | a3b | a2b | e | a3 | a2 |
| a2b | a2b | a2 | ab | b | a3b | a | e | a3 |
| a3b | a3b | a3 | a2b | ab | b | a2 | a | e |
Notice the cycle e, a, a2, a3. It can be seen from the multiplication table that successive powers of a behave this way. The reverse is also true. In other words: (a3)2 = a2, (a3)3 = a, and (a3)4 = e. This behavior is true for any cycle in any group – a cycle may be traversed in either direction.
Cycles that contain a non-prime number of elements implicitly have cycles that are not shown in the graph. For the group Dih4 above, we might want to draw a line between a2 and e since (a2)2 = e, but since a2 is part of a larger cycle, this is not done.
There can be ambiguity when two cycles share an element that is not the identity element. Consider for example, the simple quaternion group, whose cycle graph is shown on the right. Each of the elements in the middle row when multiplied by itself gives −1 (where 1 is the identity element). In this case we may use different colors to keep track of the cycles, although symmetry considerations will work as well.
As noted earlier, the two edges of a 2-element cycle are typically represented as a single line.
The inverse of an element can be identified in the cycle graph in this fashion: It is the element whose distance from the identity is the same if going through the cycle in the opposite direction.
History
Cycle graphs were investigated by the number theorist Daniel Shanks in the early 1950s as a tool to study multiplicative groups of residue classes.[2] Shanks first published the idea in the 1962 first edition of his book Solved and Unsolved Problems in Number Theory.[3] In the book, Shanks investigates which groups have isomorphic cycle graphs and when a cycle graph is planar.[4] In the 1978 second edition, Shanks reflects on his research on class groups and the development of the baby-step giant-step method:[5]
The cycle graphs have proved to be useful when working with finite Abelian groups; and I have used them frequently in finding my way around an intricate structure [77, p. 852], in obtaining a wanted multiplicative relation [78, p. 426], or in isolating some wanted subgroup [79].
Cycle graphs are used as a pedagogical tool in Nathan Carter's 2009 introductory textbook Visual Group Theory.[6]
Graph characteristics of particular group families
Certain group types give typical graphs:
Cyclic groups Zn, order n, is a single cycle graphed simply as an n-sided polygon with the elements at the vertices:
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| Z1 | Z2 = Dih1 | Z3 | Z4 | Z5 | Z6=Z3×Z2 | Z7 | Z8 |
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| Z9 | Z10=Z5×Z2 | Z11 | Z12=Z4×Z3 | Z13 | Z14=Z7×Z2 | Z15=Z5×Z3 | Z16 |
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| Z17 | Z18=Z9×Z2 | Z19 | Z20=Z5×Z4 | Z21=Z7×Z3 | Z22=Z11×Z2 | Z23 | Z24=Z8×Z3 |
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| Z2 | Z22 = Dih2 | Z23 = Dih2×Dih1 | Z24 = Dih22 |
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When n is a prime number, groups of the form (Zn)m will have (nm − 1)/(n − 1) n-element cycles sharing the identity element:
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| Z22 = Dih2 | Z23 = Dih2×Dih1 | Z24 = Dih22 | Z32 |
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Dihedral groups Dihn, order 2n consists of an n-element cycle and n 2-element cycles:
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| Dih1 = Z2 | Dih2 = Z22 | Dih3 | Dih4 | Dih5 | Dih6=Dih3×Z2 | Dih7 | Dih8 | Dih9 | Dih10=Dih5×Z2 |
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Dicyclic groups, Dicn = Q4n, order 4n:
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| Dic2 = Q8 | Dic3 = Q12 | Dic4 = Q16 | Dic5 = Q20 | Dic6 = Q24 |
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Other direct products:
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| Z4×Z2 | Z4×Z22 | Z6×Z2 | Z8×Z2 | Z42 |
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Symmetric groups – The symmetric group Sn contains, for any group of order n, a subgroup isomorphic to that group. Thus the cycle graph of every group of order n will be found in the cycle graph of Sn.
See example: Subgroups of S4
Example: Subgroups of the full octahedral group
The full octahedral group is the cross product of the symmetric group S4 and the cyclic group Z2.
Its order is 48, and it has subgroups of every order that divides 48.
In the examples below nodes that are related to each other are placed next to each other,
so these are not the simplest possible cycle graphs for these groups (like those on the right).
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| S4 × Z2 (order 48) | A4 × Z2 (order 24) | Dih4 × Z2 (order 16) | S3 × Z2 = Dih6 (order 12) |
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| S4 (order 24) | A4 (order 12) | Dih4 (order 8) | S3 = Dih3 (order 6) |
Like all graphs a cycle graph can be represented in different ways to emphasize different properties. The two representations of the cycle graph of S4 are an example of that.
  The cycle graph of S4 shown above emphasizes the three Dih4 subgroups. |
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See also
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Wikimedia Commons has media related to Group cycle graphs. |
External links
References
- ↑ Sarah Perkins (2000). "Commuting Involution Graphs for A˜n, Section 2.2, p.3, first figure" (PDF). Birkbeck College, Malet Street, London, WC1E 7HX: School of Economics, Mathematics and Statistics. Retrieved 2016-01-31.
- ↑ Shanks 1978, p. 246.
- ↑ Shanks 1978, p. xii.
- ↑ Shanks 1978, pp. 83–98, 206–208.
- ↑ Shanks 1978, p. 225.
- ↑ Carter, Nathan (2009), Visual Group Theory, Classroom Resource Materials, Mathematical Association of America, ISBN 978-0-88385-757-1
- Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica 1990, §4.2.3 Cycles, Stars, and Wheels, pp. 144-147
- Shanks, Daniel (1978) [1962], Solved and Unsolved Problems in Number Theory (2nd ed.), New York: Chelsea Publishing Company, ISBN 0-8284-0297-3
- Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematics Cambridge, England: Cambridge University Press, 2003, §6.2.4 Cycles, Stars, and Wheels pp. 248-249