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The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families.
These groups are characterized by i) an n-fold proper rotation axis C n; ii) n 2-fold proper rotation axes C 2 normal to C n; iii) a mirror plane σ h normal to C n and containing the C 2 s. The D 1h group is the same as the C 2v group in the pyramidal groups section. The D 8h table reflects the 2007 discovery of errors in older references. [4]
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Dih n: the dihedral group of order 2n (often the notation D n or D 2n is used) K 4: the Klein four-group of order 4, same as Z 2 × Z 2 and Dih 2; D 2n: the dihedral group of order 2n, the same as Dih n (notation used in section List of small non-abelian groups) S n: the symmetric group of degree n, containing the n! permutations of n elements ...
Thus, the Cayley table of a group is an example of a latin square. An alternative and more succinct proof follows from the cancellation property. This property implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one-to-one map. The result follows from the fact that one-to-one maps on finite sets are ...
General linear group, denoted by GL(n, F), is the group of n-by-n invertible matrices, where the elements of the matrices are taken from a field F such as the real numbers or the complex numbers. Group representation (not to be confused with the presentation of a group). A group representation is a homomorphism from a group to a general linear ...
The center of the general linear group over a field F, GL n (F), is the collection of scalar matrices, { sI n ∣ s ∈ F \ {0} }. The center of the orthogonal group, O n (F) is {I n, −I n}. The center of the special orthogonal group, SO(n) is the whole group when n = 2, and otherwise {I n, −I n} when n is even, and trivial when n is odd.
These groups are usually classified by some typical normal subgroup, this normal subgroup is denoted by G 0 and are written in the third column of the table. The notation 2 1+4 − stands for the extraspecial group of minus type of order 32 (i.e. the extraspecial group of order 32 with an odd number (namely one) of quaternion factor).