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Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. [1] [2] For an abelian group, each conjugacy class is a set containing one element (singleton set).
In 1912 Dehn gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2 (the genus 0 and genus 1 cases being trivial). It is known that the conjugacy problem is undecidable for many classes of groups. Classes of group ...
S 6 has exactly one (class) of outer automorphisms: Out(S 6) = C 2. To see this, observe that there are only two conjugacy classes of S 6 of size 15: the transpositions and those of class 2 3. Each element of Aut(S 6) either preserves each of these conjugacy classes, or exchanges them. Any representative of the outer automorphism constructed ...
Thus the conjugacy class within the Euclidean group E(n) of a translation is the set of all translations by the same distance. The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of all translations. So, this is the conjugate closure of a singleton containing a translation.
By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. Cl(g) = {g}. The center is the intersection of all the centralizers of elements of G: = (). As centralizers are subgroups, this again shows that the center is a subgroup.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
Note that χ ρ is constant on conjugacy classes, that is, χ ρ (π) = χ ρ (σ −1 πσ) for all permutations σ. Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KS n is semisimple. In these cases the irreducible ...
The finite group is abelian if and only if =.; One has = #where () is the number of conjugacy classes of .. If is not abelian then () / (this result is sometimes called the 5/8 theorem [5]) and this upper bound is sharp: there are infinitely many finite groups such that () = /, the smallest one being the dihedral group of order 8.