P-part of a group element of finite order
From Encyclopedia of Mathematics
-component of a group element of finite order
Let be an element of a group , and let be of finite order. Let be a prime number. Then there is a unique decomposition such that is a -element, i.e. the order of is a power of , and is a -element, i.e. the order of is prime to . The factor is called the -part or -component of and is the -part or -component. If the order of is , , , then , .
There is an analogous -element, -element decomposition, where is some set of prime numbers. This is, of course, a multiplicatively written variant of if .
References
[a1] | B. Huppert, "Endliche Gruppen I" , Springer (1967) pp. 588; Hifsatz 19.6 |
[a2] | M. Suzuki, "Group theory I" , Springer (1982) pp. 102 |
How to Cite This Entry:
P-part of a group element of finite order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-part_of_a_group_element_of_finite_order&oldid=51555
P-part of a group element of finite order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-part_of_a_group_element_of_finite_order&oldid=51555
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article