Theory of Groups of Finite Order
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Hence, in this case, ^ (P) will be a factor of 9(P).
If now P' is any sub-group of P, then it has been seen (§ 46) that the number of independent generating operations of P' is equal to or less than p. Hence 0{P') is equal to or is a factor of ^(P).
177. Theorem IX. If a group G, of order N, is trans- formed into itself by an operation S, whose order is relatively prime to N^ (G), every operation of G is permutable with S*.
Let Pj"' be the highest power of a prime p^ which divides N.
1 Frobeniu...s, loc. cit. p. 1030.
* Frobenius, loc. cit. p. 1030.
252 ORDERS OF THE ISOMORPHISMS [17^ The number of sub-groups of 0, and therefore also of {,S^, (}\ whose order is pi«', is a factor of N; and since the order of 's is relatively prime to N, one at least of them, say P^, must be transformed into itself by S. Now the order of S is relatively prime both to p^ and to ^ (Pj) ; and therefore the isomorphism of P given on transforming its operations by S must be the identical isomorphism. In other words, 8 is permutable with every operation of Pj.
What to read after Theory of Groups of Finite Order?
You can find similar books in the "Read Also" column, or choose other free books by Burnside, William, 1852-1927 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Hence, in this case, ^ (P) will be a factor of 9(P).
If now P' is any sub-group of P, then it has been seen (§ 46) that the number of independent generating operations of P' is equal to or less than p. Hence 0{P') is equal to or is a factor of ^(P).
177. Theorem IX. If a group G, of order N, is trans- formed into itself by an operation S, whose order is relatively prime to N^ (G), every operation of G is permutable with S*.
Let Pj"' be the highest power of a prime p^ which divides N.
1 Frobeniu...s, loc. cit. p. 1030.
* Frobenius, loc. cit. p. 1030.
252 ORDERS OF THE ISOMORPHISMS [17^ The number of sub-groups of 0, and therefore also of {,S^, (}\ whose order is pi«', is a factor of N; and since the order of 's is relatively prime to N, one at least of them, say P^, must be transformed into itself by S. Now the order of S is relatively prime both to p^ and to ^ (Pj) ; and therefore the isomorphism of P given on transforming its operations by S must be the identical isomorphism. In other words, 8 is permutable with every operation of Pj.
What to read after Theory of Groups of Finite Order?
You can find similar books in the "Read Also" column, or choose other free books by Burnside, William, 1852-1927 to read onlineMoreLess
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