28 Chapter 1. Groups I The function L(n) is called Landau’s function. No general formula for L(n) is known, although Landau, in 1903, found its asymptotic behavior: lim n→∞ log L(n) n log n = 1. See Miller, The maximal order of an element of a finite symmetric group, Amer. Math. Monthly 94 (1987), 315–322. Section 1.4. Lagrange’s Theorem A subgroup H of a group G is a group contained in G such that h, h H implies that the product hh in H is the same as the product hh in G. Note that the multiplicative group H = {±1} is not a subgroup of the additive group Z, for the product of 1 and −1 in H is −1 while the “product” in Z is their sum, 0. The formal definition of subgroup is more convenient to use. Definition. A subset H of a group G is a subgroup if (i) 1 H, (ii) H is closed that is, if x, y H, then xy H, (iii) if x H, then x−1 H. Observe that G and {1} are always subgroups of a group G, where {1} denotes the subset consisting of the single element 1. A subgroup H G is called a proper subgroup a subgroup H = {1} is called a nontrivial subgroup. Proposition 1.32. Every subgroup H of a group G is itself a group. Proof. Property (ii) shows that H is closed, for x, y H implies xy H. Asso- ciativity (xy)z = x(yz) holds for all x, y, z G, and it holds, in particular, for all x, y, z H. Finally, (i) gives the identity, and (iii) gives inverses. For Galois, groups were subgroups of symmetric groups. Cayley, in 1854, was the first to define an “abstract” group, mentioning associativity, inverses, and iden- tity explicitly. He then proved Theorem 1.95: every abstract group with n elements is isomorphic to a subgroup of Sn (we discuss isomorphism in the next section). It is easier to check that a subset H of a group G is a subgroup (and hence that it is a group in its own right) than to verify the group axioms for H: associativity is inherited from G, and hence it need not be verified again. Example 1.33. (i) The set of four permutations, V = (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) , is a subgroup of S4 : (1) V α2 = (1) for each α V, and so α−1 = α V the product of any two distinct permutations in V−{(1)} is the third one. It follows from Proposition 1.32 that V is a group it is called the four-group (V abbreviates the original German term Vierergruppe).
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