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A stable [[Cohomology operation|cohomology operation]]  $  {\mathcal P}  ^ {i} $, 
$  i \geq  0 $, 
of the type  $  ( \mathbf Z _ {p} , \mathbf Z _ {p} ) $, 
where  $  p $
is a fixed odd prime number, which is the analogue modulo  $  p $
of the [[Steenrod square|Steenrod square]], and which is a homomorphism

$$ 
{\mathcal P}  ^ {i} :  H  ^ {n} ( X, Y;  \mathbf Z _ {p} )  \rightarrow  H  ^ {n+2i}( p- 1) ( X, Y;  \mathbf Z _ {p} ),
$$

defined for every pair of topological spaces  $  ( X, Y) $
and any integer  $  n $. 
The Steenrod reduced powers possess the following properties (apart from naturality  $  f ^ { * } {\mathcal P}  ^ {i} = {\mathcal P}  ^ {i} f ^ { * } $
and stability  $  \delta {\mathcal P}  ^ {i} = {\mathcal P}  ^ {i} \delta $, 
where  $  \delta :  H  ^ {q} ( Y;  \mathbf Z _ {p} ) \rightarrow H  ^ {q+1} ( X, Y;  \mathbf Z _ {p} ) $
is the coboundary homomorphism):

1)  $  {\mathcal P}  ^ {0} =  \mathop{\rm id} $;

2) if  $  2i =  \mathop{\rm dim}  x $, 
then  $  {\mathcal P}  ^ {i} x = x  ^ {p} $;

3) if  $  2i >  \mathop{\rm dim}  x $, 
then  $  {\mathcal P}  ^ {i} x = 0 $;

4) (Cartan's formula)  $  {\mathcal P}  ^ {i} ( x, y) = \sum_{j=0}^ {i} ( {\mathcal P}  ^ {i} x) \cdot ( {\mathcal P}  ^ {i-j} x) $;

5) (Adem's relation)

$$ 
{\mathcal P}  ^ {a} {\mathcal P}  ^ {b}  =  \sum _ { t= } 0 ^ { [ }  v/p] (- 1)  ^ {a+t} \left ( \begin{array}{c}
( p- 1)( b- t)- 1 \\
 a- pt 
\end{array}
 \right ) _ {p} {\mathcal P}  ^ {a+b-t}
$$

if  $  a < pb $,

$$ 
{\mathcal P}  ^ {a} \beta {\mathcal P}  ^ {b}  = \ 
\sum_{t=0}^ { [ }  a/p] (- 1)  ^ {a+t} \left ( \begin{array}{c}
( p- 1)( b- t) \\
 a- pt 
\end{array}
 \right
) _ {p} \beta {\mathcal P}  ^ {a+b-t} {\mathcal P}  ^ {t} +
$$

$$ 
+ 
\sum _ { t= } 0 ^ { [( }  a- 1)/p] (- 1)  ^ {a+t-1} \left (
\begin{array}{c}
( p- 1)( b- t)- 1 \\
 a- pt- 1 
\end{array}
 \right ) _ {p} {\mathcal P}  ^ {a+b-t} \beta {\mathcal P}  ^ {t}
$$

if  $  a \leq  pb $, 
where  $  \beta $
is the Bockstein homomorphism associated with the short exact sequence of coefficient groups  $  0 \rightarrow \mathbf Z _ {p} \rightarrow \mathbf Z _ {p  ^ {2}  } \rightarrow \mathbf Z _ {p} \rightarrow 0 $, 
while  $  ( \cdot ) _ {p} $
are the binomial coefficients reduced modulo  $  p $.

These properties are analogous to the corresponding properties of Steenrod squares, whereby the operation  $  Sq  ^ {2i} $
corresponds to the operation  $  {\mathcal P}  ^ {i} $. 
Just as for Steenrod squares, the multiplication in 4) can be considered to be both exterior ( $  \times $-
multiplication) and interior ( $  \cup $-
multiplication). Steenrod reduced powers commute with [[Suspension|suspension]] and [[Transgression|transgression]].

The properties 1)–3) uniquely characterize  $  {\mathcal P}  ^ {i} $, 
and can be constructed in the same way as Steenrod squares using the minimal acyclic free chain  $  \mathbf Z _ {p} $-
complex  $  W $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,   D.B.A. Epstein,   "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  ''Matematika'' , '''5''' :  2  (1961)  pp. 3–11; 11–30; 30–49; 50–102</TD></TR></table>

====Comments====
For more references see [[Steenrod algebra|Steenrod algebra]].