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[parent] octic group (Example)

The octic group also known as the $ 4^{th}$ dihedral group, is a non-Abelian group with eight elements. It is traditionally denoted by $ D_4$. This group is defined by the presentation

$\displaystyle < s,t \mid s^4 = t^2 = e, st = ts^{-1}>$
or, equivalently, defined by the multiplication table
$ \cdot$ $ e$ $ s$ $ s^2$ $ s^3$ $ t$ $ ts$ $ ts^2$ $ ts^3$
$ e$ $ e$ $ s$ $ s^2$ $ s^3$ $ t$ $ ts$ $ ts^2$ $ ts^3$
$ s$ $ s$ $ s^2$ $ s^3$ $ e$ $ ts^3$ $ t$ $ ts$ $ ts^2$
$ s^2$ $ s^2$ $ s^3$ $ e$ $ s$ $ ts^2$ $ ts^3$ $ t$ $ ts$
$ s^3$ $ s^3$ $ e$ $ s$ $ s^2$ $ ts$ $ ts^2$ $ ts^3$ $ t$
$ t$ $ t$ $ ts$ $ ts^2$ $ ts^3$ $ e$ $ s$ $ s^2$ $ s^3$
$ ts$ $ ts$ $ ts^2$ $ ts^3$ $ t$ $ s^3$ $ e$ $ s$ $ s^2$
$ ts^2$ $ ts^2$ $ ts^3$ $ t$ $ ts$ $ s^2$ $ s^3$ $ e$ $ s$
$ ts^3$ $ ts^3$ $ t$ $ ts$ $ ts^2$ $ s$ $ s^2$ $ s^3$ $ e$
where we have put each product $ xy$ into row $ x$ and column $ y$. The lattice of the subgroups is given below:
\begin{figure}\centerline{ \xymatrix{ & & D_4 \ & \{ e, s^2, t, ts^2 \} \ar@{-... ... <1>\ar@{-}[ull]\ar@{-}[ul]\ar@{-}[u]\ar@{-}[ur]\ar@{-}[urr]\ } } \end{figure}
where $ <a_1,\ldots,a_n>$ denotes the subgroup generated by $ a_1,\ldots ,a_n$ and $ \{b_1,\ldots ,b_n\}$ denotes the subgroup. Of those subgroups, the following are its proper normal subgroup: $ \{e,s^2,t,ts^2\}$, $ <s>$, $ \{e,s^2,st,ts\}$, and $ <s^2>$. In addition the center and commutator subgroup of the octic group is $ <s^2>$. It can also be shown that the automorphism of the octic group ( $ \operatorname{Aut}(D_4)$) is isomorphic to itself($ D_4$).[PJ] An additional property is that the subgroup of the general linear group of dimension 2 over the real numbers generated by:
$\displaystyle \left[ \begin{array}{cc} 0 & 1 \ -1 & 0 \end{array}\right],\left[ \begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right]$
is isomorphic to the octic group.

References

PJ
Pedersen, John: Groups of small order. http://www.math.usf.edu/˜eclark/algctlg/small_groups.html



"octic group" is owned by Daume.
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Other names:  $D_4$

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Cross-references: generated by, real numbers, dimension, general linear group, property, isomorphic, automorphism, commutator subgroup, center, addition, normal subgroup, subgroup generated by, subgroups, lattice, column, row, product, multiplication, presentation, group, non-abelian group, dihedral group
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This is version 2 of octic group, born on 2004-11-05, modified 2004-11-05.
Object id is 6453, canonical name is OcticGroup.
Accessed 3001 times total.

Classification:
AMS MSC: 20F55 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Reflection and Coxeter groups)
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