Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Definition (original) (raw)
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Algebra for Cryptologists, 2016
Groups, Rings and Ideals An algebraic structure generally consists of a set, and one or more binary operations on that set, as well as a number of properties that the binary operation(s) has (have) to satisfy. In the following pages we shall discuss the most important algebraic structures, viz groups, rings and fields. For the purposes of our applications, viz applications to Cryptology, it will be sufficient if in all such structures we restrict ourselves to the commutative instances, i.e. we look only at groups and rings in which the binary operations are commutative. Thus where we discuss groups we shall only consider the so-called "Abelian groups", and where we discuss rings, we limit ourselves to those which are in the literature referred to as (unsurprisingly) "commutative rings". More precisely they are "commutative rings with identity", since even the existence of an identity element is not a requirement for rings in general. 3.1 Groups Definitions A group is a pair fG; g consisting of a set G and a binary operation on G, such that the following are satisfied: 1. 8a; b; c 2 G; a .b c/ D .a b/ c (i.e. the operation is associative). 2. 9e 2 G such that 8a 2 G; e a D a e D a. e is called the identity element of the group. 3. 8a 2 G 9x 2 G such that x a D a x D e. x is called the inverse of a and denoted by a 1 (in our present choice of notation). If, in addition to the above, the binary operation satisfies 4. 8a; b 2 G; a b D b a, i.e. the operation is commutative, then the structure is called an Abelian group. Before giving any examples, we note a few immediate consequences of this definition: The identity element is unique. For suppose e and e 0 are both identity elements. Then e D e e 0 D e 0 ;
isara solutions, 2012
In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.
An Elementary Introduction to Groups and Representations
2000
The set of complex numbers with absolute value one (i.e., of the form e iθ) forms a group under complex multiplication. This group is commutative. This group is the unit circle, denoted S 1. 2.7. Invertible matrices. For each positive integer n, the set of all n × n invertible matrices with real entries forms a group with respect to the operation of matrix multiplication. This group in non-commutative, for n ≥ 2. We check closure: the product of two invertible matrices is invertible, since (AB) −1 = B −1 A −1. Matrix multiplication is associative; the identity matrix (with ones down the diagonal, and zeros elsewhere) is the identity element; by definition, an invertible matrix has an inverse. Simple examples show that the group is noncommutative, except in the trivial case n = 1. (See Exercise 8.) This group is called the general linear group (over the reals), and is denoted GL(n; R). 2.8. Symmetric group (permutation group). The set of one-to-one, onto maps of the set {1, 2, • • • n} to itself forms a group under the operation of composition. This group is non-commutative for n ≥ 3. We check closure: the composition of two one-to-one, onto maps is again oneto-one and onto. Composition of functions is associative; the identity map (which sends 1 to 1, 2 to 2, etc.) is the identity element; a one-to-one, onto map has an inverse. Simple examples show that the group is non-commutative, as long as n is at least 3. (See Exercise 10.) This group is called the symmetric group, and is denoted S n. A one-to-one, onto map of {1, 2, • • • n} is a permutation, and so S n is also called the permutation group. The group S n has n! elements. 2.9. Integers mod n. The set {0, 1, • • • n − 1} forms a group under the operation of addition mod n. This group is commutative. Explicitly, the group operation is the following. Consider a, b ∈ {0, 1 • • • n − 1}. If a + b < n, then a + b mod n = a + b, if a + b ≥ n, then a + b mod n = a + b − n. (Since a and b are less than n, a+b−n is less than n; thus we have closure.) To show associativity, note that both (a+b mod n)+c mod n and a+(b+c mod n) mod n are equal to a + b + c, minus some multiple of n, and hence differ by a multiple of n. But since both are in the set {0, 1, • • • n − 1}, the only possible multiple on n is zero. Zero is still the identity for addition mod n. The inverse of an element a ∈ {0, 1, • • • n − 1} is n − a. (Exercise: check that n − a is in {0, 1, • • • n − 1}, and that a + (n − a) mod n = 0.) The group is commutative because ordinary addition is commutative. This group is referred to as "Z mod n," and is denoted Z n. 3. Subgroups, the Center, and Direct Products Definition 1.7. A subgroup of a group G is a subset H of G with the following properties: 1. The identity is an element of H. 2. If h ∈ H, then h −1 ∈ H. 3. If h 1 , h 2 ∈ H, then h 1 h 2 ∈ H .
European Journal of Pure and Applied Mathematics
A nonempty set G is a g-group [with respect to a binary operation ∗] if it satisfies the following properties: (g1) a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G; (g2) for each a ∈ G, there exists an element e ∈ G such that a ∗ e = a = e ∗ a (e is called an identity element of a); and, (g3) for each a ∈ G, there exists an element b ∈ G such that a ∗ b = e = b ∗ a for some identity element eof a. In this study, we gave some important properties of g-subgroups, homomorphism of g-groups, andthe zero element. We also presented a couple of ways to construct g-groups and g-subgroups.
Groups formed by redefining multiplication
Canadian Mathematical Bulletin, 1988
Let G be a group with elements 1,…, n such that the group operation agrees with ordinary multiplication whenever the ordinary product of two elements lies in G. We show that if n is odd, then G is abelian.
Some Sufficient conditions for a group to be abelian
TURKISH JOURNAL OF MATHEMATICS
A group is said to satisfy a word w in the symbols {x, x −1 , y, y −1 } provided that if the 'x' and 'y' are replaced by arbitrary elements of the group then the equation w = 1 is satisfied. This paper studies certain equations in words, as above, which together with other conditions imply that groups which satisfy these equations and conditions must be abelian.
Group theory for Maths, Physics and Chemistry students
The operation is associative, i.e., for all g, h, k ∈ G we have g • (h • k) = (g • h) • k. 2. G contains an identity element, i.e., an element e that satisfies e • g = g • e = g for all g ∈ G. 3. Each element of G has an inverse, i.e., for each g ∈ G there is an h ∈ G such that g • h = h • g = e. This element is denoted by g −1. The cardinality of G is called the order of the group, and often denoted by |G|; it may be infinite. If the operation is not only associative but commutative as well (meaning g • h = h • g for all g, h ∈ G), then G is called an Abelian group. 1.2. BASIC NOTIONS Exercise 1.2.2. Prove that a group G cannot have more than one identity. Also, the notation g −1 for the inverse of g seems to indicate the uniqueness of that inverse; prove this. Let us give some examples of groups. Example 1.2.3. 1. All sets of transformations found in §1.1 form groups; in each case, the composition • serves as operation •. Some of them are Abelian; like the the chair, and some aren't, like those of the cube. 1. α(e, m) = m for all m ∈ M , and 2. α(g, α(h, m)) = α(gh, m) for all m ∈ M and g, h ∈ G.