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Difference between revisions of "Binary tetrahedral group"

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(Start article: Binary tetrahedral group)
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#REDIRECT [[Tetrahedral space]]
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{{TEX|done}}{{MSC|20}}
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The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as:
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$$
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\langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ .
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$$
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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.
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This group may be realised as the group of invertible [[Hurwitz number]]s:
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$$
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\pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ .
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$$
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The group has an action on the three-sphere with [[tetrahedral space]] as quotient.
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==References==
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<table>
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<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>
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</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