2014 Putnam B5

In the 75th annual Putnam Games, participants compete at mathematical games.
Patniss and Keeta play a game in which they take turns choosing an element
from the group of invertible $n \times n$ matrices with entries in the field
$\mathbb{Z}/p \mathbb{Z}$ of integers modulo $p$, where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are:

(1)
A player cannot choose an element that has been chosen by either player on any previous turn.

(2)
A player can only choose an element that commutes with all previously chosen elements.

(3)
A player who cannot choose an element on his/her turn loses the game.

Patniss takes the first turn. Which player has a winning strategy?

(Your answer may depend on $n$ and $p$.)

在第 75 届年度普特南运动会上，参与者进行数学游戏竞赛。

帕特尼斯和基塔玩一个游戏，轮流选择一个元素

来自可逆 $n \times n$ 矩阵组，其中包含字段中的条目

$\mathbb{Z}/p \mathbb{Z}$ 以 $p$ 为模的整数，其中 $n$ 是固定正整数，$p$ 是固定质数。游戏规则是：

(1)

玩家不能选择任何一个玩家在上一回合中选择过的元素。

(2)

玩家只能选择与所有先前选择的元素可交换的元素。

(3)

无法在回合中选择元素的玩家将输掉游戏。

帕特尼斯先行。哪位玩家有获胜策略？
(您的答案可能取决于 $n$ 和 $p$。)