Which particular overlap are you looking at, group theory?

你体在看哪个重叠部分，群论？

But some of the group theory in there is a little bit difficult.

但其的一些群论有点难。

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

实际上， 大多数解魔方的人正是这样做的，他们甚至使用一种群论符号来表示旋转。

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

然而，借助群论，我们析魔方，并确定一系列排列组合，从而达成解谜的目标。

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

这些置换群称为S子n，它们在群论扮演着重要角色。

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

这是许多不同例子的一个，理解这些单群的性质实际上在群论之有重要意义。

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.

好吧， 这是关于怪兽群的规模， 但要解释这意味着什么，我们需要先退一步谈谈群论。

I understand, you know, GR, I understand mathematics, I understand some group theory, fiber bone, I can get a little of it, the age theory.

我知道，比如广义相对论（GR）， 我懂数学，懂一些群论，纤维丛理论，我理解其一部分， 还有弦理论。

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.

这个问题乍看之下可有些费解，但若您愿意耐心听我解释，群论这一抽象的数学领域将揭示其答案。

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.

当一个对应关系对于所有乘积都保持为真时，这被称为“同构” ， 这可是群论最重要的概念。

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.

在你的数学生涯， 你会看到某个群会从看似无关的情境出现，随着这种情况的增多， 你会更好地理解群论的本质。

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?

为了说明这为什么有用，回想一下我们说过群论可以用来证明五次多项式方程没有公式解， 就像二次方程那样？

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， 数学家约翰·麦凯转而研究群论的一个相邻领域，他发现一个与196,883非常相似的数字出现在了一个完全不相关的背景，或者至少是几乎不相关。