Schur multiplier for product of groups.

Question

In the book The Schur multiplier by Karpilovsky, Theorem 2.2.10 says ,

Let $G_1$ and $G_2$ be two finite groups then the Schur multiplier $$M(G_1\times G_2)=M(G_1)\times M(G_2)\times (G_1\otimes G_2).$$

But I was studying a research article https://arxiv.org/abs/1804.11308, in which it is given that $$M(G_1\times G_2)=M(G_1)\times M(G_2)\times (\frac{G_1}{G_1'}\otimes \frac{G_2}{G_2'}).$$

My question is that which result I should follow?

Answer 1

As suggested by @Derek Holt sir, the book "The Schur multiplier" was written a in $1987$ and the result mentioned in the book was proved by Schur in $1907$ whether the notion of non abelian tensor square was given by Brown and Loudy again in $1987$. Therefore second result should be considered in general.