By Thomas C. Craven
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Extra resources for Abstract algebra [Lecture notes]
151. Let F be a field, R a nonzero ring and f : F → R a surjective homomorphism. We claim f is actually an isomorphism. 11) that its kernel is zero. But its kernel must be an ideal in F and a field has only two ideals: (0) and F . The kernel can’t be all of F , for then the image is just the zero ring. Therefore the kernel must be zero. Exercise 5, p. 151. If I is an ideal in an integral domain R, then R/I need no longer be an integral domain. Indeed, a simple example of this is R = Z and I = (6).
Actually, this describes a semigroup. To make it more interesting, we require that inverses always exist. Definition, p. 163. A group is a nonempty set G with a binary operation ∗ that satisfies (1) Closure: if a, b ∈ G, then a ∗ b ∈ G. (2) Associativity: a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G. (3) There exists an element e ∈ G satisfying a ∗ e = e ∗ a = a for all a ∈ G. e is called the identity element. (4) For each a ∈ G, there exists an element b ∈ G satisfying a ∗ b = b ∗ a = e. b is called the inverse of a.
Much of our work will be with finite groups; that is, one with only finitely many elements. The number of elements in G is called the order of G, denoted |G|. If G is infinite, we say it has infinite order. We typically use whatever notation is convenient for the operation ∗. We often write it as + if the group is abelian, because it then behaves like addition in a ring. In fact, any ring is a group if we consider only its + operation. If the group is nonabelian, a more common notation is the one we use for multiplication: ab for a ∗ b.