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By Euler L.

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Example text

A cyclic group) is abelian (resp. cyclic). A refinement of a tower G = Go => GI => • . => Gm is a tower which can be obtained by inserting a finite number of subgroups in the given tower. e. Gm = {e} in the above notation). 1. Let G be a finite group. An abelian tower of G admits a cyclic refinement. Let G be a finite solvable group. Then G admits a cyclic tower whose last element is {e}. Proof The second assertion is an immediate consequence of the first, and it clearly suffices to prove that if G is finite, abelian, then G admits a cyclic tower ending with {e}.

If n ~ 5 then the alternating group An is simple. Proof. Let N be a non-trivial normal subgroup of An' We prove that N contains some 3-cycle , whence the theorem follows by (b). Let 0" EN , 0" id, be an element which has the maximal number of fixed points; that is , integers i such that O"(i) = i. It will suffice to prove that 0" is a 3-cycle or the identity. Then some orbits have more than one element. Suppose all orbits have 2 elements (except for the fixed points) . Since 0" is even, there are at least two such orbits.

7 . In general, an operation of G on S is said to be transitive if there is only one orbit. Examples. The symmetric group Sn operates transitively on {I, 2, . . , n}. (See p. 1 of Chapter VII, we shall see a non-trivial example of transitive action of a Galois group operating on the primes lying above a given prime in the ground ring. In topology, suppose we have a universal covering space p : X' -. X , where X is connected. Given x E X , the fundamental group 7l'1 (X) operates transitively on the inverse image p -l (x).

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An observation on the sums of divisors by Euler L.


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