By Paul Alexandroff, Mathematics, Hazel Perfect, G.M. Petersen
Beginning with introductory examples of the gang inspiration, the textual content advances to concerns of teams of variations, isomorphism, cyclic subgroups, basic teams of hobbies, invariant subgroups, and partitioning of teams. An appendix offers easy thoughts from set idea. A wealth of straightforward examples, basically geometrical, illustrate the first innovations. routines on the finish of every bankruptcy offer extra reinforcement.
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Extra resources for An Introduction to the Theory of Groups
DEFINITION OF A HOMOMORPHIC MAPPING AND ITS KERNEL � 2. EXAMPLES OF HOMOMORPHIC MAPPINGS CHAPTER EIGHT. PARTITIONING OF A GROUP RELATIVE TO A GIVEN SUBGROUP. DIFFERENCE MODULES � 1. LEFT AND RIGHT COSETS 1. Left cosets 2. The ease of a finite group G 3. Right cosets 4. The coincidence of the left and right cosets in the case of an invariant subgroup 5. Examples � 2. THE DIFFERENCE MODULE CORRESPONDING TO A GIVEN INVARIANT SUBGROUP 1. Definition 2. The homomorphism theorem APPENDIX. ELEMENTARY CONCEPTS FROM THE THEORY OF SETS � 1.
GROUPS OF PERMUTATIONS � 1. DEFINITION OF A PERMUTATION GROUP � 2. THE CONCEPT OF A SUBGROUP. EXAMPLES FROM THE THEORY OF PERMUTATION GROUPS 1. Examples and definition 2. A condition for a subset of a group to be a subgroup � 3. PERMUTATIONS CONSIDERED AS MAPPINGS OF A FINITE SET ONTO ITSELF. EVEN AND ODD PERMUTATIONS 1. Permutations considered as mappings 2. Even and odd permutations CHAPTER THREE. SOME GENERAL REMARKS ABOUT GROUPS. THE CONCEPT OF ISOMORPHISM � 1. THE “ADDITIVE” AND THE “MULTIPLICATIVE” TERMINOLOGY IN GROUP THEORY � 2.
The addition table of a cyclic group of order has the form: We can interpret this addition table as a second definition of a cyclic group of order . We have investigated the case that for a given element a of the group G there exist two different whole numbers m1 and m2 with the property that m1a = m2a. We consider now the case that no two such numbers exist, so that therefore all the elements are distinct. In this case there is a one-to-one correspondence between the elements (3) and the whole numbers: To the element ma corresponds the whole number m, and conversely.
An Introduction to the Theory of Groups by Paul Alexandroff, Mathematics, Hazel Perfect, G.M. Petersen