![](https://static.youtibao.com/asksite/comm/h5/images/m_q_title.png)
[主观题]
设G是有限群,且H<G.证明:设群G是其子群G1与G2的直积,即 G=G1×G2. 证明:G/G1≌G2, G/G2≌G1.
设群G是其子群G1与G2的直积,即 G=G1×G2. 证明:G/G1≌G2, G/G2≌G1.
![](https://static.youtibao.com/asksite/comm/h5/images/m_q_a.png)
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因为G=G1×G2,故G/G1={aG1|a∈G2}.现定义:φ:G/G1→G2,aG1→a.由G1∩G2={e}知,对aG1,bG1∈G/G1(a,b∈G2)有又因为φ(aG1.bG1)=φ(abG1)=ab=φ(aG1)φ(bG1),故φ为同构映射,因此G/G1≌G2.同理可证,G/G2≌G1.
![](https://static.youtibao.com/asksite/comm/h5/images/solist_ts.png)