By Cohn P.M.

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And 2. 38. Theorem of Lagrange (Joseph Louis Lagrange, 1736 1813) Let G be a finite group. Then the order ord(a) of any element a ∈ G divides the group order |G|, or with other words a|G| = e holds for all a ∈ G. Proof. Since G is finite, every element a ∈ G has finite order: ord(a) = |aZ | = d < ∞. 3. the order of the subgroup aZ ≤ G divides |G|. 39. 1. R: Let G be a group and a, b ∈ G elements of order m, n ∈ N respectively. Show that ord(ak ) = m/ gcd(k, m) and ord(ab) = mn, if ab = ba and the numbers m, n are relatively prime.

Formally, the above sum is infinite, and as such not well defined in the framework of algebra, but since na = 0 for only finitely many a ∈ M , one can define na χa := a∈M na χa . a,na =0 Furthermore one usually writes simply a instead of χa and thinks of the elements in Z[M ] as finite “formal sums” na · a a∈M 62 with integral coefficients in the elements of M . Now a free abelian group is defined to be a group isomorphic to a group Z[M ]; so an abelian group is free iff there is a subset M (a “basis”), such that any element has a unique representation as a finite linear combination a∈M na a.

R: Let H ⊂ S4 be the subgroup consisting of the identity and the products of two disjoint 2-cycles. Show: H ⊂ S4 is a normal subgroup and S4 /H ∼ = S3 . Hint: S3 ∩ H = {id}. 5. Let Aff n (R) := {f ∈ S(Rn ); f (x) = Ax + b with A ∈ GLn (R), b ∈ Rn } be the affine linear group, cf. 3. Show: The subgroup T := {τb ; b ∈ Rn } (where τb (x) = x + b is the translation with the vector b ∈ Rn ) is normal. Determine a homomorphism σ : GLn (R) −→ Aut(Rn ), such that Aff n (R) ∼ = Rn ×σ GLn (R)! Is GLn (R) ≤ Aff n (R) a normal subgroup?