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 diversifications, isomorphism, cyclic subgroups, easy teams of pursuits, invariant subgroups, and partitioning of teams. An appendix offers ordinary recommendations from set concept. A wealth of easy examples, basically geometrical, illustrate the first ideas. workouts on the finish of every bankruptcy offer extra reinforcement.
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Additional info for An Introduction to the Theory of Groups
Prove that the group of all positive numbers (with arithmetical multiplication as the group operation) is isomorphic to the group of all real numbers (with arithmetical addition as the group operation). ) 6. If a group is isomorphic to one of its proper subgroups, what is its order? 7. Find a group of permutations on four numbers which is isomorphic (a) to the group of rotations of a square, and (b) to Klein’s four-group. 8. Find a group of permutations on six numbers which is isomorphic to the symmetric group S3.
We have defined the expression na for arbitrary positive and for arbitrary negative integral n. Finally we agree to write 0a = 0 (where 0 on the left-hand side denotes the number zero and 0 on the right-hand side denotes the null element of the group). Now let p and q be any two whole numbers. From our definition it follows that We obtain the following result: The set H(a) of the elements of a group G, which can be represented in the form na for integral n, form a group with respect to the law of addition defined in the group G.
4. * We can convince ourselves of this by investigating the ten subsets of the group S3, which contain the element P0 and consist of four elements, as well as the five subsets which contain P0 and consist of five elements. But the non-existence of subgroups of S3 of orders 4 and 5 follows at once from the following general theorem which will be proved later (Chapter VIII): The order of every subgroup H of a finite group G is a divisor of the order of the group G. * The reader to whom this paragraph presents difficulties may omit it at a first reading, and need only come back to it just before Chapter VI.
An Introduction to the Theory of Groups by Paul Alexandroff, Mathematics, Hazel Perfect, G.M. Petersen