By David B Surowski

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The kernel of the homomorphism F → D determined by x → n, y → h can be shown to be xk , y 2 , (yx)2 (we’ll prove this below). Thus D has presentation D = x, y| xk = y 2 = (xy)2 = e . One need not always write each “relation” in the form r = e. Indeed, the above presentation might just as well have been written as D = x, y| xk = y 2 = e, yxy = x−1 . , without reference to a given group G. Thus, if one were to write “Consider the group G = x, y| x4 = y 2 = (xy)2 = e , ” then one means the following.

Thus F is a finite dimensional vector space over the field Fp ( alternatively denoted Z/(p), the integers, modulo p). 4. SEPARABILITY AND THE GALOIS CRITERION 57 that if n is the dimension of F over Fp , then |F| = pn . 1, F× is a cyclic group, and so the elements of F are precisely the roots of xq − x, where q = pn . In other words, F is a splitting field over Fp for the polynomial xq − x. From this we infer immediately the following. 7 Two finite fields F1 and F2 are isomorphic if and only if they have the same order.

6. THE SYMMETRIC AND ALTERNATING GROUPS 27 Show that there are two distinct decompositions of the vertices of the above graph into systems of imprimitivity: one is as four sets of two vertices each and the other is as two sets of four vertices each. In the second decomposition, if V denotes the vertices and if V = V1 ∪ V2 is the decomposition of V into two sets of imprimitivity of four vertices each, show that the setwise stabilizer of V1 is isomorphic with S4 .

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