By Helmut Hasse

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**Additional resources for Higher Algebra (2 Volumes)**

**Sample text**

1I. Groups B2A =-= 0; therefore, by (~) they are the uniquely determined solutions of those equations. Analogously to the conventions agreed upon at the end of Section 1, we write ... ,. for ... , A-lA-I, A-I, E, A, AA, ... (integral powers of A). , which is valid by Theorem 15, it then follows by the definition of the arithmetical operations in the domain of the integers that AmAn = Am+n, (Am)n = Amn for arbitrary integers -In, n. t. We formulate the following two theorems especially on account uf later application.

Analogously to the conventions agreed upon at the end of Section 1, we write ... ,. for ... , A-lA-I, A-I, E, A, AA, ... (integral powers of A). , which is valid by Theorem 15, it then follows by the definition of the arithmetical operations in the domain of the integers that AmAn = Am+n, (Am)n = Amn for arbitrary integers -In, n. t. We formulate the following two theorems especially on account uf later application. 'rhe first is merely a rephrasing of postulate (2). Theorem 16. If A is a fixed element of a group 0), then each of the products AB and BA rtms through all elements of 61, each once, if B does.

One through the angle 2n;, the other 4n 3 3 about the axis which is perpendicular to the plane of the triangle and passes through the center of the triangle; c) Three rotations B o' B I , B 2, each through the angle :t, about one of the three medians of the triangle. In setting up this set the rotation sense submitted in b) as well as the rotation axes specified in c) may be regarded as fixed in space, that is, not subjected to the rotations of the triangle. Proceeding from a fixed initial position these rotations can be illustrated by their end positions as follows: ~ , b~~~~b 3'~ '~3~t~, 3 Now, if multiplication in eM is defined as the successive application of the rotations in question, then eM il:.