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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200103.png" />-component of a group element of finite order''
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''$p$-component of a group element of finite order''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200104.png" /> be an element of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200105.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200106.png" /> be of finite order. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200107.png" /> be a [[Prime number|prime number]]. Then there is a unique decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200108.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p1200109.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001011.png" />-element, i.e. the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001012.png" /> is a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001014.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001016.png" />-element, i.e. the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001017.png" /> is prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001018.png" />. The factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001019.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001021.png" />-part or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001023.png" />-component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001025.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001027.png" />-part or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001029.png" />-component. If the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001035.png" />.
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Let $x$ be an element of a [[Group|group]] $G$, and let $x$ be of finite order. Let $p$ be a [[Prime number|prime number]]. Then there is a unique decomposition $x=yz=zy$ such that $y$ is a $p$-element, i.e. the order of $y$ is a power of $p$, and $z$ is a $p'$-element, i.e. the order of $z$ is prime to $p$. The factor $y$ is called the $p$-part or $p$-component of $x$ and $z$ is the $p'$-part or $p'$-component. If the order of $x$ is $r=p^{\alpha}s$, $(p,s)=1$, $bp^{\alpha}+cs=1$, then $y=x^{sc}$, $z=x^{p^{a_{b}}}$.
  
There is an analogous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001037.png" />-element, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001039.png" />-element decomposition, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001040.png" /> is some set of prime numbers. This is, of course, a multiplicatively written variant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001041.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120010/p12001042.png" />.
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There is an analogous $\pi$-element, $\pi'$-element decomposition, where $\pi$ is some set of prime numbers. This is, of course, a multiplicatively written variant of $\textbf{Z}/(nm)\simeq\textbf{Z}/(n)\times\textbf{Z}/(m)$ if $(n,m)=1$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen I" , Springer  (1967)  pp. 588; Hifsatz 19.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Suzuki,  "Group theory I" , Springer  (1982)  pp. 102</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen I" , Springer  (1967)  pp. 588; Hifsatz 19.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Suzuki,  "Group theory I" , Springer  (1982)  pp. 102</TD></TR></table>

Revision as of 19:09, 6 February 2021

$p$-component of a group element of finite order

Let $x$ be an element of a group $G$, and let $x$ be of finite order. Let $p$ be a prime number. Then there is a unique decomposition $x=yz=zy$ such that $y$ is a $p$-element, i.e. the order of $y$ is a power of $p$, and $z$ is a $p'$-element, i.e. the order of $z$ is prime to $p$. The factor $y$ is called the $p$-part or $p$-component of $x$ and $z$ is the $p'$-part or $p'$-component. If the order of $x$ is $r=p^{\alpha}s$, $(p,s)=1$, $bp^{\alpha}+cs=1$, then $y=x^{sc}$, $z=x^{p^{a_{b}}}$.

There is an analogous $\pi$-element, $\pi'$-element decomposition, where $\pi$ is some set of prime numbers. This is, of course, a multiplicatively written variant of $\textbf{Z}/(nm)\simeq\textbf{Z}/(n)\times\textbf{Z}/(m)$ if $(n,m)=1$.

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