Center of unitriangular matrix group:UT(3,p)
This article is about a particular subgroup in a group parametrized by a prime (so we get one group for each prime and a corresponding subgroup for it), up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (generically) group of prime order and the group is (generically) prime-cube order group:U(3,p) (see subgroup structure of prime-cube order group:U(3,p)).
The subgroup is a normal subgroup and the quotient group is (generically) elementary abelian group of prime-square order.
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Contents
Definition
Let be an odd prime (The case , where we get center of dihedral group:D8, is somewhat different, though many things are similar).
The group is the group of unipotent upper-triangular matrices over the prime field of elements, and is also the unique (up to isomorphism) non-abelian group of order and exponent . It is defined by the presentation:
.
In the matrix description, each matrix can be described by the three entries . The matrix looks like:
The multiplication of matrices and gives the matrix where:
With the matrix description, we can set as the matrix with and the other two entries zero, as the matrix with and the other two entries zero, and as the matrix with and the other two entries zero.
Now define the subgroup:
In other words, this is the subgroup given by .
Cosets
The subgroup has cosets each of size , where each coset is parametrized by fixed values of . In other words, for any fixed and , the corresponding coset of in is the set of all matrices with those values of and , and with varying over all of .
Complements
The subgroup has no permutable complements and also no lattice complements.
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
complemented normal subgroup | normal subgroup with a complement | No | nilpotent and non-abelian implies center is not complemented |
permutably complemented subgroup | subgroup with a permutable complement. | No | nilpotent and non-abelian implies center is not complemented |
lattice-complemented subgroup | subgroup with a lattice complement. | No | nilpotent and non-abelian implies center is not complemented |
Effect of subgroup operators
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | the whole group | -- | prime-cube order group:U(3,p) |
centralizer | the whole group | current page | prime-cube order group:U(3,p) |
normal core | the subgroup itself | current page | group of prime order |
normal closure | the subgroup itself | current page | group of prime order |
characteristic core | the subgroup itself | current page | group of prime order |
characteristic closure | the subgroup itself | current page | group of prime order |
commutator with whole group | the trivial subgroup | current page | trivial group |
Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
center | set of elements that commute with every element | To see that every element of is in the center, simply use the formula for multiplication and simplify algebraically. The same algebraic approach can be used to show that nothing else is in the center. |
derived subgroup | subgroup generated by commutators of all pairs of elements in the group, smallest subgroup with abelian quotient | The quotient (called the abelianization) is , which is isomorphic to elementary abelian group of prime-square order. No smaller subgroup works, because the quotient by the trivial subgroup is isomorphic to , which is non-abelian. |
socle | subgroup generated by all the minimal normal subgroups | It is the unique minimal normal subgroup (hence is also a monolith). In general, for a finite -group, the socle is . |
Frattini subgroup | intersection of all the maximal subgroups | is the intersection of maximal subgroups obtained by taking along with any single element outside it. |
ZJ-subgroup | center of the join of abelian subgroups of maximum order | The join of abelian subgroups of maximum order (the Thompson subgroup) is the whole group, so its center is . |
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | center is normal |
characteristic subgroup | invariant under all automorphisms | Yes | center is characteristic, derived subgroup is characteristic |
fully invariant subgroup | invariant under all endomorphisms | Yes | derived subgroup is fully invariant |
verbal subgroup | generated by set of words | Yes | derived subgroup is verbal |
normal-isomorph-free subgroup | no other isomorphic normal subgroup | Yes | |
isomorph-free subgroup, isomorph-containing subgroup | No other isomorphic subgroups | No | There are other subgroups of order . In fact, every element of the group has order . |
isomorph-normal subgroup | Every isomorphic subgroup is normal | No | There are other subgroups of order two that are not normal: etc. |
homomorph-containing subgroup | contains all homomorphic images | No | There are other subgroups of order . |