Binomial Coefficient involving Power of Prime

Lemma

Let $p$ be a prime number, and let $k \in \Z$.

Then:

$\displaystyle \binom {p^n k} {p^n} \equiv k \pmod p$

where $\displaystyle \binom {p^n k} {p^n}$ is a binomial coefficient.

Proof

From Prime Power of Sum Modulo Prime we have:

$(1) \qquad \left({a + b}\right)^{p^n} \equiv \left({a^{p^n} + b^{p^n}}\right) \pmod p$

We can write this:

$\left({a + b}\right)^{p^n k} = \left({\left({a + b}\right)^{p^n}}\right)^k$

By $(1)$ and Congruence of Powers, we therefore have:

$\left({a + b}\right)^{p^n k} \equiv \left({a^{p^n} + b^{p^n}}\right)^k \pmod p$

The coefficient $\displaystyle \binom {p^n k} {p^n}$ is the binomial coefficient of $b^{p^n}$ in $\left({a + b}\right)^{p^n k} = \left({\left({a + b}\right)^{p^n}}\right)^k$.

Expanding this using the Binomial Theorem, we find that the coefficient of $b^{p^n}$ is $k$.

So:

$\displaystyle \binom {p^n k} {p^n} \equiv k \pmod p$

$\blacksquare$

Comment

This lemma is used in the proof of the Sylow Theorems.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.6$: Example $42 \ (5)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 11$: Lemma $11.2$