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The orders of products and commutators in prime-power groups

Published online by Cambridge University Press:  24 October 2008

T. E. Easterfield
T. E. Easterfield
Affiliation:
Clare CollegeCambridge
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Extract

It is known † that, if G is any p-group (group whose order is a power of the prime p) the class of which is less than p, then the order of a product of elements of G cannot exceed the orders of all the factors. This result in general ceases to hold when the class of G exceeds p − 1. In the present paper, we prove the following general results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 1 , January 1940 , pp. 14 - 26
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

† Hall, , Proc. London Math. Soc. (2), 36 (1933), 73–77.Google Scholar

† Z 1 is the central of G; and, for i> 1, Z i/Z i−1 is the central of G/Z i−1.

† Hall, , Proc. London Math. Soc. (2), 40 (1936), 477.Google Scholar If H is a subgroup of G, [G:H] denotes the index of H in G.

† Hall, , Proc. London Math. Soc. (2), 40 (1936), 485.Google Scholar