This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and its possible extensions. In this paper, it is shown that the PQ penny flip game can be associated, in a precise way, with the dihedral group $D_8$ and that within $D_8$ there exist precisely two classes of equivalent winning strategies for Q. This is achieved by proving that there are exactly two different sequences of states that can guarantee Q’s win with probability $1.0$. It is demonstrated that the game can be played in every dihedral group $D_{8n}$, where $n\ge 1$, without any significant change. A formal examination of what happens when Q can draw their moves from the entire $U\left(2\right)$, leads to the conclusion that, again, there are exactly two classes of winning strategies for Q, each class containing an infinite number of equivalent strategies, but all of them sending the coin through the same sequence of states as before. Finally, when general extensions of the game, with the quantum player having $U\left(2\right)$ at their disposal, are considered, a necessary and sufficient condition for Q to surely win against Picard is established: Q must make both the first and the last move in the game.

中文翻译：

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PQ Penny Flip游戏和二面体之间的联系

本文的灵感来自于PQ一分钱翻转游戏。它采用小组理论的概念来研究原始游戏及其可能的扩展。本文表明，PQ一分钱翻转游戏可以与二面体组精确关联$d_8$ 还有那个 $d_8$ Q恰好存在两类等效的获胜策略。这可以通过证明存在两种完全不同的状态序列来保证Q获胜的可能性来实现。 $\mathrm{1个}。0$。事实证明，该游戏可以在每个二面体组中进行$d_{8ñ}$， 在哪里 $ñ\ge \mathrm{1个}$，没有任何重大变化。正式检查当Q可以从整体上汲取它们的举动时会发生什么$ü（\mathrm{2个}）$得出的结论是，同样，对于Q，恰好有两类获胜策略，每类包含无限数量的等效策略，但是所有这些都通过与以前相同的状态序列发送代币。最后，当对游戏进行一般扩展时，$ü（\mathrm{2个}）$ 在他们的支配下，为Q肯定要击败Picard确立了必要和充分的条件：Q必须在游戏中进行第一和最后一步。