
gair-abelian samooh

ganit mein, aur vishesh roop seey samooh siddhant mein, ack gair-abelian samooh, jise kabhi-kabhi gair-commutative samooh kahaa jaataa haye, ack samooh ("g", ≤) haye jisamen tatvon kii cumm seey cumm ack jodee "a" aur "g" kii "b" maujood hoti haye, jaise kii "a" ≤ "b" ≤ "a". samoohon kaa yeh varg abelian samoohon key vipareet haye. (ack abelian samooh mein, samooh tatvon key sabhi jode awagaman karate hain). 
gair-abelian samooh ganit aur bhautiki mein vyaapak hain. gair-abelian samooh key sabse saral udaharanon mein seey ack crum 6 kaa dihedral samooh haye. yeh sabse chhota seemit gair-abelian samooh haye. bhautiki seey ack samanya udaaharan ghoornan samooh haye isaliye (3) teen aayamon mein (udaaharan key liye, ack aksh key saath kuchh 90 degree ghoomana aur phir ack alag aksh key saath 90 degree ghoomana unhein vipareet crum mein karane key samaan naheen haye).
donon asatat samooh aur nirantar samooh gair-abelian hoe sakate hain. adhikansh dilchasp jhooth samooh gair-abelian hain, aur yeye gaze siddhant mein ack mahatvapurna bhoomika nibhaate hain.