Difference between revisions of "Binary tetrahedral group"
From Encyclopedia of Mathematics
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| + | The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as: | ||
| + | $$ | ||
| + | \langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ . | ||
| + | $$ | ||
| + | It is finite of order 24. It has the [[alternating group]] $A_4$ as quotient by the centre and the [[quaternion group]] of order 8 as a quotient. | ||
| + | |||
| + | This group may be realised as the group of invertible [[Hurwitz number]]s: | ||
| + | $$ | ||
| + | \pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ . | ||
| + | $$ | ||
| + | |||
| + | The group has an action on the three-sphere with [[tetrahedral space]] as quotient. | ||
| + | |||
| + | ==References== | ||
| + | <table> | ||
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 76 ISBN 0-521-20125-X {{ZBL|0732.51002}}</TD></TR> | ||
| + | </table> | ||
Revision as of 15:34, 19 January 2021
2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]
The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as: $$ \langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ . $$ It is finite of order 24. It has the alternating group $A_4$ as quotient by the centre and the quaternion group of order 8 as a quotient.
This group may be realised as the group of invertible Hurwitz numbers: $$ \pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ . $$
The group has an action on the three-sphere with tetrahedral space as quotient.
References
| [1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 76 ISBN 0-521-20125-X Zbl 0732.51002 |
How to Cite This Entry:
Binary tetrahedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_tetrahedral_group&oldid=51423
Binary tetrahedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_tetrahedral_group&oldid=51423