Group Theory I EssentialsResearch & Education Assoc., 2013 M01 1 - 112 pages REA’s Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Group Theory I includes sets and mapping, groupoids and semi-groups, groups, isomorphisms and homomorphisms, cyclic groups, the Sylow theorems, and finite p-groups. |
Contents
| 1 | |
GROUPOIDS AND SEMIGROUPS | 21 |
GROUPS | 30 |
ISOMORPHISMS AND HOMOMORPHISMS | 52 |
CYCLIC GROUPS COSETS | 61 |
HOMOMORPHISMS | 77 |
THE SYLOW THEOREMS | 82 |
FINITE pGROUPS | 90 |
Other editions - View all
Common terms and phrases
a₁ abelian group additive group associative law binary operation C₂x called CARTESIAN PRODUCT complex numbers cosets of H denoted elements of G equivalence classes equivalence relation EXAMPLE Let exists finite cyclic group finite group G and H G order G=HXK G₁ G₂ gp(x group G group of integers group of isometries group of order group of prime groupoid H in G H-conjugate h₁ h₂ Hence identity of G integers inverse element isometries isomorphic kerf kernel Let G let H multiplication table normal subgroup number of distinct number of inversions odd permutation one-to-one order of G order p³ ordered pair p-group positive integer proper subgroup quasi-group semi-group subgroup of G subgroup of order subgroups H subsets of G Sylow p-subgroups Sylow Theorem symmetric group symmetry t₂ Theorem of Lagrange Theorem on Homomorphisms unit element x₁ y₁ ал
Popular passages
Page 12 - The cartesian product of the sets A and B is the set of all ordered pairs (a, b) where ae A and be B.
Page 71 - SUBGROUPS A subgroup, H, of a group, G, is said to be a normal (or invariant) subgroup of G if and only if for every element ae G.
Page iii - It condenses the vast amount of detail characteristic of the subject matter and summarizes the essentials of the field. It will thus save hours of study and preparation time. The book provides quick access to the important facts, principles, theorems, concepts, and equations in the field.
Page 5 - Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers Even if we can list the elements of a set, it may not be practical to do so.
Page 30 - DEFINITION OF A GROUP A group G is a set of elements and a binary operation which we will call "product
Page 36 - Thm.X.4.17 below gives as a special case the familiar result, (RS)-1 = (S-1)(R-1), that the inverse of a product is the product of the inverses in reverse order.
Page 37 - A subset H of a group G is said to be a subgroup of G if H is itself a group with respect to the operation of composition defined in G.
Page 10 - A u (B n C) = (A u B) n (A u C) A n (B u C) = (A n B) u (A n C) 8. De Morgan's laws : (A u B)' = A' n B' and (A n B)' = A' u B
Page 37 - A nonempty subset// of G is a subgroup of G if and only if...