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=13865
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=13865
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article