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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-2-25\">习题 2.25</a> \n>\n> 设 $V$ 是组成法则为 $+$ 的阿贝尔群，$G$ 是在 $V$ 上有左作用的任意群，仍记作 $g:v\\mapsto gv$。进一步假设每个 $g$ 的作用都是 $V$ 的同态：\n>\n> $$\n> g(v+w)=gv+gw.\n> $$\n>\n> (a) 证明 $G\\times V$ 在以下组成法则下是群：\n>\n> $$\n> (g,v)(g',v')=(gg',v+gv').\n> $$\n>\n> 此群称为 $G$ 与 $V$ 的<u id=\"concept-2-semi-direct-product\" class=\"concept-term\" data-concept-identity=\"urn:uuid:c65e599c-ab76-4124-9f5a-0f1f9997e585\" data-concept-route=\"/concept-identities#concept-c65e599c-ab76-4124-9f5a-0f1f9997e585\">半直积(semi-direct product)</u>，记作 $G\\text{\\textcircled{S}}V$。\n>\n> (b) 证明 $(g,0)$ 型元素组成一个同构于 $G$ 的子群；$V$ 同构于子群 $(e,V)$，且后者是正规子群。\n>\n> (c) 证明半直积的每个元素都有唯一分解 $vg$，其中 $g\\equiv(g,0)\\in G$，$v\\equiv(e,v)\\in V$。",
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