Math 561 H Fall 2011
نویسنده
چکیده
2. Let G be a set with binary operation (a, b) 7→ ab and consider the following possible axioms: (1) ∀ a, b ∈ G, a(bc) = (ab)c. (2) ∃ e ∈ G,∀ a ∈ G, ae = ea = a. (3) ∀ a ∈ G,∃ b ∈ G, ab = ba = e. (3’) ∀ a ∈ G,∃ b ∈ G, ab = e. Prove that the axioms (3) and (3’) are equivalent. That is, show that (1), (2), and (3) hold if and only if (1), (2), and (3’) hold. (One direction is easy. For the other direction, let a ∈ G. Then there exist b, c ∈ G such that ab = e and bc = e. Show that a = c.)
منابع مشابه
Math 561 H Fall 2011 Homework 3 Solutions Drew
Proof. Suppose that φ; Z/nZ → Z/nZ is an automorphism with φ(1) = a. By the homomorphism property we have φ(x) = φ(1 + · · · + 1) = φ(1) + · · · + φ(1) = a + · · · + a = ax. Thus the image of φ is the (additive) cyclic subgroup 〈a〉 ≤ Z/nZ. Then since φ is surjective we must have 〈a〉 = Z/nZ. By the Lemma, this happens if and only if a and n are coprime, i.e. a ∈ (Z/nZ)×, in which case the map φ(...
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تاریخ انتشار 2011