Search results
Results from the WOW.Com Content Network
A (Z)-group is a group faithfully represented as a doubly transitive permutation group in which no non-identity element fixes more than two points. A (ZT)-group is a (Z)-group that is of odd degree and not a Frobenius group , that is a Zassenhaus group of odd degree, also known as one of the groups PSL(2,2 k +1 ) or Sz(2 2 k +1 ) , for k any ...
List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
Given a group and a subgroup , and a fixed element , one can consider the corresponding left coset: := {:} .Cosets are a natural class of subsets of a group; for example consider the abelian group of integers, with operation defined by the usual addition, and the subgroup of even integers.
An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; [note 3] an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n.
The quotient group, G / Z(G), is isomorphic to the inner automorphism group, Inn(G). A group G is abelian if and only if Z(G) = G. At the other extreme, a group is said to be centerless if Z(G) is trivial; i.e., consists only of the identity element. The elements of the center are central elements.
A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group (Z / 2Z) n (of cardinality 2 n) acts faithfully on a set of size 2n. This is not always the case, for example the cyclic group Z / 2 n Z cannot act faithfully on a set of size less than 2 n.
Examples of groups that do not have property (T) include The additive groups of integers Z, of real numbers R and of p-adic numbers Q p. The special linear groups SL(2, Z) and SL(2, R), as a result of the existence of complementary series representations near the trivial representation, although SL(2,Z) has property (τ) with respect to ...
The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function (sequence A002322 in the OEIS). In other words, λ ( n ) {\displaystyle \lambda (n)} is the smallest number such that for each a coprime to n , a λ ( n ) ≡ 1 ( mod n ) {\displaystyle a^{\lambda (n)}\equiv 1 ...