# Reference request concerning splitting fields for groups that are related to special symmetric groups

Denote the symmetric group of order $$n!$$ by $$S_n$$. Let $$H:=S_p$$ for an odd prime $$p$$.

Every finite field $$k$$ is a splitting field for $$kH$$, in particular $$k:=\mathbb{F}_p$$.

Questions:

Is $$k:=\mathbb{F}_p$$ also a splitting field for $$kG$$ where

a) $$G:=H\times H$$ ?

b) $$G:=H \wr C_2$$ ?

I would be interested in references appearing in the literature which deal with these (similar) questions.

Thank you in advance for the help.

Edit: $$k$$ is a splitting field of $$S_n$$ if the $$k$$-algebra $$kS_n$$ splits over $$k$$, i.e. if for every simple $$kS_n$$-module $$M$$, we have End$$_{kSn}(M)\cong k$$. (cf. Splitting fields of symmetric groups)

Remark: I looked at https://ncatlab.org/nlab/show/direct+product+group and due to remark 2.2 my questions might not have an affirmative answer for arbitrary groups, but I was interested, if the statement is nevertheless true in these special cases.

• $\Bbb{F}_p$ is not a splitting field of $S_p$: groupprops.subwiki.org/wiki/Splitting_field Commented Aug 23, 2020 at 1:38
• Thank you for the comment. I edited the question. Commented Aug 23, 2020 at 2:03

This condition you call being a splitting field is really just saying that all of the irreducible $$\overline{\mathbb{F}}_p$$ representations are defined over $$\mathbb{F}_p$$. For symmetric groups this is a standard fact, and can be found say in James' book.

Since the irreducible representations of $$S_n$$ are all defined over $$\mathbb{F}_p$$, so are the tensor products $$V \otimes W$$ which are the irreducible representations of $$S_n \times S_n$$.

For wreath products $$S_n \wr C_2$$ the irreducible representations in characteristic $$p \ne 2$$ are basically constructed the same way as characteristic zero: Each irreducible $$S_k$$ representation $$D_\lambda$$ can be extended in two ways to irreducible representations $$D_\lambda^0$$ and $$D_\lambda^1$$ of $$S_k \wr C_2$$ by declaring that each $$C_2$$ acts either trivially or by a sign. A general irreducible representation of $$S_n \wr C_2$$ is of the form $$Ind_{S_k \wr C_2 \times S_{n-k} \wr C_2} (D_\lambda^0 \otimes D_\mu^1)$$. Clearly these are defined over $$\mathbb{F}_p$$ since the $$D_\lambda$$'s are.

This is not an accurate answer but would be helpful. Take a look at chapter 2 of

Fields and Galois Theory By J.S. Milne that is available freely

and a quick review and lecture of KEITH CONRAD. There are some books on splitting fields like

Rotman, Joseph, Galois theory., Universitext. New York, NY: Springer. xiv, 157 p. (1998). ZBL0924.12001.

or

Lidl, Rudolf; Niederreiter, Harald, Introduction to finite fields and their applications., Cambridge: Univ. Press,. xi, 416 p. (1994). ZBL0820.11072.

or chapter 9 of

Roman, Steven, Field theory, Graduate Texts in Mathematics 158. New York, NY: Springer (ISBN 0-387-27677-7/hbk). xii, 332 p. (2006). ZBL1172.12001.

• Thank you very much. Commented Sep 9, 2020 at 17:10