逐字来源、作者主张与候选译文分别保留。工程核验不批准 OCR、译文或数学正确性。
群同态 · 已登记definition
沿用现有原著来源登记及角色。
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
两个群$G$和$G'$,映射$\varphi : G \rightarrow G'$保持群乘法，
$$
\varphi(a b)=\varphi(a) \varphi(b)
$$
则称这个映射是<u class="concept-term">同态(homomorphism)</u>
群同态 · 已登记definition
沿用现有原著来源登记及角色。
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
Let $G$ and $G ^ { \prime }$ be groups. A homomorphism $\varphi : G \to G ^ { \prime }$ is a map from $G$ to $G ^ { \prime }$ that preserves products,
$$
\varphi (a b) = \varphi (a) \varphi (b) \quad \text { for all } a, b \in G.
$$
阿贝尔群定义与加法记号
交换的群称为阿贝尔群；采用加法记号时默认交换。
条件与约定：Szekeres 先讨论一般群，再明确加法记号的交换约定。
符号：保留原文符号。
逐字作者来源与版本
A group $G$ is called abelian if the law of composition is commutative,
$$
g h = h g \quad \text { for all } g, h \in G.
$$
The notation gh for the product of two elements is the default notation. Other possibilities are $a \cdot b , a \times b , a + b , a \circ b$ , etc. When the law of composition is written as an addition $g + h$ , we will always assume that the commutative law holds, $g + h = h + g$ . In this case the identity element is usually written as 0, so that (Gp2) reads $g + 0 = 0 + g = g$ . The inverse is then written $-g$, with (Gp3) reading $g + ( - g ) = 0$ or, more simply, $g - g = 0$ Again, the associative law means we never have to worry about parentheses in expressions such as $a + b + c + \cdots + f .$
子群条件
子群含母群单位元，对乘积及取逆封闭。
条件与约定：Szekeres 允许非阿贝尔母群。
符号：保留原文符号。
逐字作者来源与版本
A subgroup $H$ of a group $G$ is a subset that is a group in its own right. A subset $H \subseteq G$ is thus a subgroup ifit contains the identity element of $G$ and is closed under the operations of taking products and inverses:
(a) $h$, $k \in H \Rightarrow h k \in H$ (closure with respect to taking products);
(b) the identity $e \in H ;$
(c) $h \in H \Rightarrow h ^ { - 1 } \in H$ (closure with respect to taking inverses).
一般群的商群构造
正规子群的陪集在诱导乘法下形成商群。
条件与约定：Szekeres 要求 正规；不是任意子群都可形成群商。
符号：保留原文符号。
逐字作者来源与版本
When we multiply left cosets of a subgroup $H$ together, for example
$$
g H g ^ {\prime} H = \{g h g ^ {\prime} h ^ {\prime} \mid h, h ^ {\prime} \in H \},
$$
the result is not in general another coset. On the other hand, the product ofcosets ofa norma subgroup $N$ is always another coset,
$$
g N g ^ {\prime} N = g g ^ {\prime} N N = (g g ^ {\prime}) N,
$$
and satisfies the associative law,
$$
(g N g ^ {\prime} N) g ^ {\prime \prime} N = (g g ^ {\prime} g ^ {\prime \prime}) N = g N (g ^ {\prime} N g ^ {\prime \prime} N).
$$
Furthermore, the coset $e N = N$ plays the role of an identity element, while every coset has an inverse $( g N ) ^ { - 1 } = g ^ { - 1 } N$ . Hence the cosets of a normal subgroup $N$ form a group called the factor group of $G$ by $N$, denoted $G / N$