Special groups of low order

## Special groups of low order

In this chapter we shall give tables which give explicit information about several interesting examples of groups which have order less than $100$. For the most part we shall not prove the results given in these tables. We stronly suggest that, in each individual case, the reader do the appropriate computations to check the information in these tables, for it is exactly in the computations in examples such as these that the subject of group theory "comes alive".

Let us begin with a list of the diffeent groups of order $\le 15$. The reader should think that about extending this table to include all groups of order, say, $\le 100$. The beautiful book [CM] may be very helpful for such a project.

Note also that the finite abelian groups are completely determined by the Fundamental Theorem of Abeleian groups, Theorem [Insert Reference].

 In the following table: $Q$ denotes the quaternion group. ${C}_{k}$ denotes the cyclic group of order $k$. ${D}_{k}$ denotes the dihedral group of order $2k$. ${S}_{k}$ denotes the symmetric group on $k$ letters. ${A}_{k}$ denotes the alternating group on $k$ letters.

Group Order Abelian
$⟨1⟩$ $1$ Yes
${C}_{2}$ $2$ Yes
${C}_{3}$ $3$ Yes
${C}_{4}$ $4$ Yes
${C}_{2} × {C}_{2}$ $4$ Yes
${C}_{5}$ $5$ Yes
${C}_{6}\cong {C}_{2} × {C}_{3}$ $6$ Yes
${S}_{3}\cong {D}_{3}$ $6$ No
${C}_{7}$ $7$ Yes
${C}_{8}$ $8$ Yes
${C}_{2} × {C}_{4}$ $8$ Yes
${C}_{2} × {C}_{2} × {C}_{2}$ $8$ Yes
${D}_{4}$ $8$ No
$Q$ $8$ No
${C}_{9}$ $9$ Yes
${C}_{3} × {C}_{3}$ $9$ Yes
${C}_{10}\cong {C}_{2} × {C}_{5}$ $10$ Yes
${D}_{5}$ $10$ No
${C}_{12}\cong {C}_{3} × {C}_{4}$ $12$ Yes
${C}_{2} × {C}_{2} × {C}_{3}$ $12$ Yes
${D}_{6}\cong {C}_{2} × {D}_{3}$ $12$ Yes
${A}_{4}$ $12$ No
$⟨S,T\mid {S}^{3}={T}^{2}={\left(ST\right)}^{2}⟩$ $12$ No
${C}_{13}$ $13$ Yes
${C}_{14}\cong {C}_{2} × {C}_{7}$ $14$ Yes
${D}_{7}$ $14$ Yes
${C}_{15}\cong {C}_{3} × {C}_{5}$ $15$ Yes

## References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)