Mizar theorem GROUP_6:93
Victor Makarov viktmak _AT_ gmail.com <[email protected]> Fri, 8 Dec 2023 08:53:12 -0500
| Newsgroups | gmane.comp.mathematics.mizar |
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Dear Forum: I am trying to understand the statement of the theorem GROUP_6:93: https://mx10.uwb.edu.pl/fmlurlsvc/?fewReq=:B:JVk/NDA9PyNzODcrNSNsYTg1ND81NCN2bGJrZHFwd2A4PWM1MzJjZDRjNmBmNzQ0ZmA9NmRkNzM0M2RkPWBjYDFmYGE3ZmcxNCNxODQyNTc1MTYwPDEjdGxhODZHPUF3QFBdNTc2MDY2KDZHPUF3QFBcNTc2MDY2I3dmdXE4aGx/ZHcoY2p3cGhFaGx/ZHcrcHJnK2BhcCt1aSNmODY8I21haTg1&url=http%3a%2f%2fmizar.org%2fJFM%2fVol3%2fgroup_6.abs.html%23K2 theorem :: GROUP_6:93 for N being strict normal Subgroup of G holds (B "\/" N)./.(B "\/" N,N)`*`, B./.(B /\ N) are_isomorphic; What does mean (B "\/" N)" and (B "\/" N,N)`*` ? How is the theorem GROUP_6:93 related to the Second Isomorphism Theorem? Second Isomorphism Theorem: (Theorem B in Wikipedia): https://mx10.uwb.edu.pl/fmlurlsvc/?fewReq=:B:JVk/NDA9PyNzODcrNSNsYTg1ND81NCN2bGJrZHFwd2A4MWEwYTYyMTZkMz02NzIzZDdmM2Y0ZmBkPDAxPGQ0N2AyNjI3NTIxMSNxODQyNTc1MTYwPDEjdGxhODZHPUF3QFBdNTc2MDY2KDZHPUF3QFBcNTc2MDY2I3dmdXE4aGx/ZHcoY2p3cGhFaGx/ZHcrcHJnK2BhcCt1aSNmODY8I21haTg1&url=https%3a%2f%2fen.wikipedia.org%2fwiki%2fIsomorphism_theorems Theorem B (groups) Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then the following hold: The product SN is a subgroup of G, The subgroup N is a normal subgroup of SN}, The intersection S/\N is a normal subgroup of S, and The quotient groups (SN)/N and S/(S/\N)} are isomorphic. (Wikipedia uses another symbol(\cap) for the intersection). Thanks in advance, Victor Makarov