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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 ''a''6 = ''e''. Then the set of powers of ''a''2, is a cycle, but this is really no new information. Similarly, ''a''5 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 ''a''2, ..., ''a''''n''−1 to ''a''''n'', etc., until ''e'' is reached. The result is the cycle graph. When ''a''2 = ''e'', ''a'' has order 2 (is an involution), and is connected to ''e'' by two edges. It is conventional to show only one edge in this case. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cycle graph (algebra)」の詳細全文を読む スポンサード リンク
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