CAUTION: The subgroup is not the special linear group of degree two! In fact, the special linear group coincides with the whole general linear group.Īs the general affine group:GA(1,q) where Note that it is a feature of field:F2 that semisimple elements form a multiplicative subgroup - they do not form a multiplicative subgroup for larger field sizes. The subgroup comprising the semisimple elements, although the ones of order three are not diagonalizable over field:F2 and can be diagonalized only over field:F4. The cyclic part comprising rotations, i.e., orientation-preserving elements.Īs the general linear group of degree two over field:F2 The subgroup is the unique maximal normal subgroup - the group is a one-headed groupĭescription in alternative interpretations of the whole group Interpretation ofĪs the dihedral group of degree three, order six Intersection of all maximal normal subgroups (same reason as 3-Sylow core order has two prime factors 2 and 3) The subgroup is the unique normal 3-Sylow subgroup Largest normal subgroup whose order is a power of 3 normal core of any 3-Sylow subgroup The subgroup is the unique nontrivial abelian normal subgroup The subgroup is the uniqueminimal normal subgroup (i.e., monolith) - the group is a monolithic group It is the derived subgroup and its commutator with the whole group equals itself.
Groupspro 3 series#
Subgroup at which the lower central series of the whole group stabilizes We can also explicitly compute all commutators - these are precisely the identity element and the two 3-cycles. The quotient is cyclic group:Z2, which is abelian no other subgroup has abelian quotient. Subgroup generated by commutators of all pairs of group elements, smallest subgroup with abelian quotient The subgroup is a characteristic subgroup of the whole group and arises as a result of many common subgroup-defining functions on the whole group. Number of conjugacy classes in automorphism class Normal subgroup having a normal complement Normal subgroup having a permutable complementĬharacteristic subgroup having a permutable complement Groups Properties related to complementation Property VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groupsįor information on these as subgroups, see S2 in S3. The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2. The subgroup is (up to isomorphism) cyclic group:Z3 and the group is (up to isomorphism) symmetric group:S3 (see subgroup structure of symmetric group:S3). This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). Assign contacts to groups (single and fast.
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