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By Pierre Deligne

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The converse is also true: if the inversion is an automorphism, the group is abelian. e. (ba)−1 = (ab)−1 , and ab = ba. Hence the correspondence α : a → a−1 is an automorphism if, and only if, the group is abelian. If α ∈ Aut(G) and H ≤ G, then α(H) is again a subgroup of G and is isomorphic to H. 17. A subgroup H of a group G is a characteristic subgroups if every automorphism of G maps H to itself. 10. 1. The identity subgroup and the group itself are characteristic subgroups. 2. The group of integers has a unique nonidentity automorphism, and this takes n to −n.

9) If G1 = G, ϕ is an endomorphism. The set of endomorphisms of G is denoted End(G). If ϕ is surjective we say that G1 is homomorphic to G. 4) G1 is isomorphic to G. We recover the notion of an isomorphism as a special case of that of a homomorphism. 8) is a homomorphism, called the canonical homomorphism. 9); moreover, ϕ(a−1 ) = ϕ(a)−1 (take b = a−1 ). Let K be the set of elements of G whose image is the identity element of G1 , K = {a ∈ G | ϕ(a) = 1}. Then K is a subgroup of G. Indeed, i) ϕ(1) = 1, so that 1 ∈ K; ii) if a ∈ K, then ϕ(a−1 ) = ϕ(a)−1 = 1, and therefore a−1 ∈ K; iii) if a, b ∈ K then ϕ(ab) = ϕ(a)ϕ(b) = 1 · 1 = 1, and thus ab ∈ K.

Pht t ; ii) if ϕ(n) is Euler’s function, then: ϕ(n) = ϕ(ph1 1 )ϕ(ph2 2 ) · · · ϕ(pht t ) and ϕ(pk ) = pk − pk−1 = pk−1 (p − 1). 16. The dihedral group Dn is isomorphic to S n only for n = 3. 17. D4 has three subgroups of order 4, one cyclic and two Klein groups, and five subgroups of order 2. 18. D6 contains two subgroups isomorphic to S 3 . ] 19. Determine a group of 2 × 2 complex matrices isomorphic to the quaternion group. ±1 k , where k = 0, 1, . . , n − 1, form a group isomorphic to 01 Dn (integers mod n).

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Catégories tannakiennes by Pierre Deligne


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