Theory And Applications of Finite Groups
Theory And Applications of Finite Groups
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This proof can easily be effected by a method employed above. It is easy to prove that the number of the non-cyclic subgroups of order p a in a group of order /> a+1 is of the form l+kp whenever this number is not zero and p>2 by showing that there must be p cyclic subgroups of order p a whenever there is one such subgroup. The number of the subgroups of order p a+l , which contain a given non-cyclic group of order p a and are themselves contained in a group of order p m , is also of the form l...+kp.
Let r a and r a+ i represent respectively the numbers of the non-cyclic subgroups of order p" and p a+l , and let s x represent the number of the subgroups of order p a+1 in which a given non-cyclic subgroup of order p a occurs while s y denotes the num- Digitized by VjOOQ IC § 52] NUMBER OF NON-CYCLIC SUBGROUPS 129 ber of non-cyclic subgroups of order p a contained in a given subgroup of order ^* +1 . We then count each subgroup of order p"* 1 as many times as it contains a non-cyclic subgroup of order p a and thus arrive at the equation 2 s x = 2 s 9 .
What to read after Theory And Applications of Finite Groups?
You can find similar books in the "Read Also" column, or choose other free books by Miller, G. A. (George Abram), 1863-1951 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
This proof can easily be effected by a method employed above. It is easy to prove that the number of the non-cyclic subgroups of order p a in a group of order /> a+1 is of the form l+kp whenever this number is not zero and p>2 by showing that there must be p cyclic subgroups of order p a whenever there is one such subgroup. The number of the subgroups of order p a+l , which contain a given non-cyclic group of order p a and are themselves contained in a group of order p m , is also of the form l...+kp.
Let r a and r a+ i represent respectively the numbers of the non-cyclic subgroups of order p" and p a+l , and let s x represent the number of the subgroups of order p a+1 in which a given non-cyclic subgroup of order p a occurs while s y denotes the num- Digitized by VjOOQ IC § 52] NUMBER OF NON-CYCLIC SUBGROUPS 129 ber of non-cyclic subgroups of order p a contained in a given subgroup of order ^* +1 . We then count each subgroup of order p"* 1 as many times as it contains a non-cyclic subgroup of order p a and thus arrive at the equation 2 s x = 2 s 9 .
What to read after Theory And Applications of Finite Groups?
You can find similar books in the "Read Also" column, or choose other free books by Miller, G. A. (George Abram), 1863-1951 to read onlineMoreLess
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