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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