By Paul Alexandroff, Mathematics, Hazel Perfect, G.M. Petersen
Beginning with introductory examples of the crowd inspiration, the textual content advances to issues of teams of diversifications, isomorphism, cyclic subgroups, easy teams of hobbies, invariant subgroups, and partitioning of teams. An appendix presents effortless suggestions from set thought. A wealth of straightforward examples, basically geometrical, illustrate the first techniques. routines on the finish of every bankruptcy supply extra reinforcement.
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Extra resources for An Introduction to the Theory of Groups
We call this subgroup the subgroup of the group G generated by the element a. � 2. Finite and infinite cyclic groups We have defined the group H(a) as the group consisting of all those elements of G which are representable in the form ma. But we have not yet considered the following question: Do two expressions m1a and m2a involving different integers m1 and m2 always give rise to two different elements of the group G, or can it happen that m1a = m2a with m1 and m2 distinct? We will concern ourselves with this problem now.
We prove that the whole group H(a) is exhausted by the elements (1), so that therefore for arbitrary integral m we have To this end we divide m by and represent m in the form where q is the quotient and r is the remainder, satisfying the condition * Then we have ma = (q + r)a = q · a + ra and since q · a = q(a) = q · 0 = 0 also ma = ra Therefore if there exist two numbers m1 and m2 such that m1a = m2a, then there exists a natural number such that the group H(a) is exhausted by the mutually distinct elements also a = 0, and more generally: The whole series simply consists of infinitely many repetitions, to the left and to the right, of the series (1).
The statement that “the element a generates the group H(a)” is equivalent to the statement that “the element a is a generating element of the group H(a)”. e. not every group is generated by a single element. Non-cyclic groups are generated not by one but by many, sometimes by infinitely many, elements. The concept of a generating element leads to the concept of a system of generators. —We consider the plane with a Cartesian system chosen in it, and denote by G the set of points P = (x, y) whose two coordinates x and y are whole numbers.