two generators, and may be traversed in either direction. More generally, the number of generators of a cycle with n elements is given by the Euler φ function of n, and any of these generators may be written as the first node in the cycle (next to the identity e); or more commonly the nodes are left unmarked. Two distinct cycles cannot intersect in a generator. Cycles that contain a non-prime number of elements have cyclic subgroups that are not shown in the graph. For the group Dih4 above, we could draw a line between a2 and e since , but since a2 is part of a larger cycle, this is not an edge of the cycle graph. There can be ambiguity when two cycles share a non-identity element. For example, the 8-element quaternion group has cycle graph shown at 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 is the node symmetric to it in its cycle, with respect to the reflection which fixes the identity. 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. Shanks first published the idea in the 1962 first edition of his book Solved and Unsolved Problems in Number Theory. In the book, Shanks investigates which groups have isomorphic cycle graphs and when a cycle graph is planar. In the 1978 second edition, Shanks reflects on his research on class groups and the development of the baby-step giant-step method: Cycle graphs are used as a pedagogical tool in Nathan Carter's 2009 introductory textbook Visual Group Theory. 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: When n is a prime number, groups of the form (Zn)m will have n-element cycles sharing the identity element: Dihedral groups Dihn, order 2n consists of an n-element cycle and n 2-element cycles: Dicyclic groups, Dicn = Q4n, order 4n: Other direct products: Symmetric