Abstract Algebra Sample Clauses

Abstract Algebra. ‌ This section provides a basic introduction to groups and finite fields. 2.1.1 Groups‌ 2.1.1. A binary operation ∗ on a set G is a function that assigns to each pair of elements a and b in G a unique a ∗ b in G. Binary operators may be of the form ∗, ·, +, ◦, ⊕, ⊗. An operation a ∗ b is said to be written in the multiplicative notation. A brinary operation written in the additive notation is denoted by a +. 2.1.2. A group (G, ∗) is a non-empty set G and a binary operation ∗ of which G1 ∗ G1 = G2 satisfies the following axioms. • Associativity: ∀a, b, c ∈ G, a ∗ (b ∗ c) = (a ∗ b) ∗ c • Neutral element: There is an element e in G such that ∀a ∈ G, e ∗ a = a ∗ e = a • Inverse element: For each a ∈ G, there is an inverse element a−1 such that a ∗ a−1 = e • Commutativity: If G is an Abelian group, then ∀a, b ∈ G, a ∗ b = b ∗ a. The order of a group G, usually denoted by |G|, is the number of elements in the set G. If G 2.1.3. A group G is cyclic if there is an element α ∈ G such that for each b ∈ G there is an integer i with b = αi. If α generates all elements of the group (G, ∗), then α is a generator of G. The order of α equals to the order of the group it generates.