假設Gamma是一個群，作用在一個集合X上。又假設phi:X 送到 {1, 2, ..., r}是一個對集合X的一個r-標號。若對任意的sigma 屬於 Gamma, sigma 不等於 id_X ，存在一個x在X裡，使得phi(x)不等於phi(sigma(x))，我們稱phi為該群作用的一個可辨識標號。若r是該群作用存在一個可辨識r-標號的最小正整數，則稱r為該群作用的『辨識數』， 記為D_{Gamma}(X)。
當給定一個圖G，G的辨識數D(G)定義成D(G) = D_{Aut(G)}(V(G))。
令S_5表示5-階對稱群。這篇論文討論S_5作用在一個集合上的辨識數。我們確定出所有S_5作用的辨識數。同時，我們也確定出所有Aut(G)=S_5的圖形G的可能的辨識數。所得到的結論證實了Albertson和Collins的一個猜測。

Suppose Gamma is a group acting on a set X. An r-labeling phi: X to {1, 2, ..., r} of X is distinguishing (with respect to the action of Gamma) if for any sigma in Gamma, sigma not equal id_X, there exists an element x in X such that phi(x) not equal phi(sigma(x)). The distinguishing number, D_{Gamma}(X), of the action of Gamma on X is the minimum r for which there is an r-labeling which is distinguishing. Given a graph G, the distinguishing number of G, D(G),is defined as D(G) = D_{Aut(G)}(V(G)). This thesis determines the distinguishing numbers of all actions of S_5. As a consequence, we
determine all the possible values of the distinguishing numbers of graphs G with Aut(G)=S_5, confirming a conjecture of Albertson and Collins.

1 Introduction 5
2 Definitions 8
3 Some useful lemmas 11
4 Actions of S5 with a single orbit 15
5 Actions with more than one orbit 32
6 Graphs G with Aut(G) = S5 40
7 Appendix: Figures illustrating the actions of S5 with one orbit 41
7.1 S5, (S5 : C5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.2 (S5, (S5 : D4)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.3 (S5, (S5 : D5)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.4 (S5, (S5 : D6)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.5 (S5, (S5 : K)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
References 91
List of Figures
1 The possible orbits of actions of S5 . . . . . . . . . . . . . . . . . 15
2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 A label-preserving permutation . . . . . . . . . . . . . . . . . . . 24
4 A label-preserving permutation . . . . . . . . . . . . . . . . . . . 24
5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8 A label-preserving permutation . . . . . . . . . . . . . . . . . . . 26
9 A label-preserving permutation . . . . . . . . . . . . . . . . . . . 27
10 A label-preserving permutation . . . . . . . . . . . . . . . . . . . 27
11 Ten permutation of the remaining five cosets . . . . . . . . . . . . 30
12 Three permutation of the remaining four cosets . . . . . . . . . . 30
13 The distinguishing number of all the subgroups of S5 . . . . . . . 31

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