P-part of a group element of finite order

$p$-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 $\mathbf{Z}/(nm)\simeq\mathbf{Z}/(n)\times\mathbf{Z}/(m)$ 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