这是用户在 2025-1-16 17:19 为 https://brilliant.org/courses/group-theory/fundamentals/subgroups/1/?from_llp=advanced-math 保存的双语快照页面，由 沉浸式翻译 提供双语支持。了解如何保存？

Subgroups  子群

In this lesson we're going to look at subgroups of a group. The idea is similar to a

A subspace is a subset of a vector space that's also a vector space itself. Specifically, it's a subset that's both closed under vector addition and scalar multiplication.

of a

A vector space is a set of vectors that is closed under vector addition and scalar multiplication.

, which you may have seen in linear algebra.
在本课中，我们将探讨群的子群。这个概念与线性代数中你可能见过的向量空间的子空间类似。

Let GG be a group. A subset H⊆GH \subseteq G is a subgroup if it forms a group under the multiplication operation already defined for G.G.
设 GG 为一个群。若子集 H⊆GH \subseteq G 在 G.G. 已定义的乘法运算下形成一个群，则称 H⊆GH \subseteq G 为 GG 的子群。

Since GG is already a group, you get some of the

A group GG is a set of elements together with a binary operation ∗* that together satisfy these axioms:

  1. Closure: For any elements xx and yy in G,G, the element x∗yx * y is also in G.G.

  2. Associativity: For any elements x,y,x, y, and zz in G,G, we have (x∗y)∗z=x∗(y∗z).(x*y)*z = x*(y*z).

  3. Identity: There is an element ee in GG satisfying e∗x=x∗e=xe*x = x*e = x for any element xx of G.G.

  4. Inverse: For every element xx of G,G, there is an element yy in GG satisfying xy=yx=e.xy = yx = e.

(like

The associative property explains that way in which numbers are grouped in addition or multiplication will not change the outcome. Therefore, (a+b)+c=a+(b+c)(a + b) + c = a + (b + c) and (a⋅b)⋅c=a⋅(b⋅c).(a \cdot b) \cdot c = a \cdot (b \cdot c).

) for free in H.H. In fact, it's not hard to see that you only need to check for

A set is closed under a binary operation when any combination of its elements by the operation is also in the set.

 under multiplication and

Given an element gg of a group with operation ∗* and identity element e,e, the inverse of gg is the unique element hh satisfying g∗h=h∗g=e.g * h = h * g = e. The inverse of gg is often denoted by g−1.g^{-1}.

: HH is a subgroup of GG if and only if for h1h_1 and h2∈H,h_2 \in H, h1h2∈Hh_1h_2 \in H and for any r∈H,r \in H, r−1∈H.r^{-1} \in H.
由于 GG 已经是一个群，你在 H.H. 中可以自动获得一些公理（如结合律）。实际上，只需检查乘法和逆的封闭性即可： HH 是 GG 的子群当且仅当对于 h1h_1 和 h2∈H,h_2 \in H, 满足 h1h2∈Hh_1h_2 \in H ，并且对于任意 r∈H,r \in H, 满足 r−1∈H.r^{-1} \in H. 。

Which of the following is not a

A subgroup is a subset of a group that is also a group under the same operation.

of

A group is cyclic when it can be generated by a single element of the group. That is, there is an element gg in the group such that any element can be written as gmg^m for some integer m.m. Every cyclic group can be identified with Zn,Z_n, the group of integers with addition modulo n.n.


以下哪个不是 Z10?Z_{10}? 的子群

Let GG be the group of functions from

A real number is a value that can represent any continuous quantity, positive or negative. Examples of real numbers include 3,−100,2.1312,13,2,3, -100, 2.1312, \frac13, \sqrt{2}, and π.\pi. The set of real numbers is often denoted by R.\mathbb{R}.

to R,\mathbb R, with group law given by addition: (f+g)(x)=f(x)+g(x).(f+g)(x) = f(x)+g(x).
设 GG 是从 R\mathbb R 到 R,\mathbb R, 的函数群，其群运算为加法： (f+g)(x)=f(x)+g(x).(f+g)(x) = f(x)+g(x).

Let H1={f∈G:f(1)=0},H_1 = \{ f \in G : f(1) = 0 \}, H2={f∈G:f(0)=1},H_2 = \{ f \in G : f(0) = 1 \}, and H3={f∈G:f(x)≥0}.H_3 = \{ f \in G : f(x) \ge 0\}.
设 H1={f∈G:f(1)=0},H_1 = \{ f \in G : f(1) = 0 \}, H2={f∈G:f(0)=1},H_2 = \{ f \in G : f(0) = 1 \}, 和 H3={f∈G:f(x)≥0}.H_3 = \{ f \in G : f(x) \ge 0\}.

Which of the HiH_i are

A subgroup is a subset of a group that is also a group under the same operation.
子群是群的一个子集，且在相同的运算下也构成一个群。

?
哪些 HiH_i 是子群？