Property of Alternating $k$-Linear Functions

Question

I have trouble understanding the proof of Lemma 3.14 from Loring Tu's An Introduction to Manifolds.

Lemma 3.14. If $f$ is an alternating $k$-linear function on a vector space $V$, then $Af = (k!)f.$

Proof. Since for alternating $f$ we have $\sigma f = (\mathrm{sgn}\,\sigma)f$, and $\mathrm{sgn}\,\sigma$ is $\pm 1$, we must have

$$ Af = \sum_{\sigma\in S_k} (\mathrm{sgn}\,\sigma) \, \sigma f = \sum_{\sigma\in S_k} (\mathrm{sgn}\,\sigma) \, (\mathrm{sgn}\,\sigma) f = (k!) f. $$

My Question

How do we get the last step with $k!$ in it from the 3rd step with two $(\mathrm{sgn}\,\sigma)$ in it?

Answer 1

For any $\sigma \in S_k$, we have that $(\operatorname{sgn}\sigma)^2 = 1$. Thus, $$\sum_{\sigma \in S_k} (\operatorname{sgn} \sigma) (\operatorname{sgn} \sigma) f = \sum_{\sigma \in S_k} f = \Big( \sum_{\sigma \in S_k}1 \Big) f$$ and since there are $k!$ possible permutations in $S_k$, in the latter we are adding $k!$ times the number $1$.