These permutation groups go by the name S sub n, and they play an important role in group theory.

这些排列群的名称为 S sub n，它们在群论中发挥着重要作用。

And, in fact, that's exactly what most solvers do, even using a group theory notation indicating turns.

而且，事实上，这正是大多求解器所做的，甚至使用指示转弯的群论符号。

However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution.

但是， 使用群论，我们可以分析立方体并确定将产生解决方案的排列序列。

This is one of many different examples where understanding the nature of these simple groups, these atoms, actually matters outside of group theory.

这是多不同例，在这些例中，理解这些简单群、这些原的本质实际上在群论外很重要。

Well, it's the size of the monster, but to explain what that means, we're gonna need to back up and talk about group theory.

嗯，这是怪物的大小， 但为了解释这意味着什么，我们需要支持并讨论群论。

That question doesn't make a lot of sense at first, but an abstract mathematical field called group theory holds the answer, if you'll bear with me.

这个问题开始并没有多大意义，但是如果你能忍受我的话，个叫做群论的抽象领域就有了答案。

When you have a correspondence where this remains true for all products, it's called an " isomorphism" , which is maybe the most important idea in group theory.

当你有个对所有产品都成立的对应关系时，它被称为“同构” ，这可能是群论中最重要的想法。

In your mathematical life, you'll see more examples of a given group arising from seemingly unrelated situations, and as you do, you'll get a better sense for what group theory is all about.

在你的生活中， 你会看到更多由看似无关的情况产生的给定群的例，当你这样做时， 你会更好地理解群论的全部内容。

To give a hint for why this is useful, remember how we said that group theory can be used to prove that there's no formula for a degree-5 polynomial the way there is for quadratic equations?

为了提示为什么它有用，还记得我们如何说过群论可以用来证明 5 次多项式不存在像二次方程那样的公式吗？

In the 1970s, mathematician John McKay was making a switch to studying group theory to an adjacent field, and he noticed that a number very similar to this 196,883 showed up in a completely unrelated context, or at least, almost.

20 世纪 70 年代， 家约翰·麦凯 (John McKay) 转而研究邻近领域的群论，他注意到个与 196,883 非常相似的字出现在完全不相关的背景中，或者至少是几乎不相关的背景中。