User:R-a-jones: Difference between revisions
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For my own persual if I need to look something up, but also may help you, pages for groups of small orders are in this table below. I intend to create more of these pages, where the order is sufficiently interesting (primes are not so interesting, neither are 2*primes, or primes^2, for example.) | For my own persual if I need to look something up, but also may help you, pages for groups of small orders are in this table below. I intend to create more of these pages, where the order is sufficiently interesting (primes are not so interesting, neither are 2*primes, or primes^2, for example.) | ||
General orders: [[Classification of groups of order a product of two distinct primes|pq]], [[Classification of groups of an order two times a prime|2p]], [[Classification of groups of order four times a prime|4p]] ([[Classification of groups of order four times a prime congruent to 1 modulo 4|p is congruent to 1 mod 4]], [[Classification of groups of order four times a prime congruent to 3 modulo 4|p is congruent to 3 mod 4]]), [[Classification of groups of prime-cube order|p^3]], [[Classification of groups of prime-fourth order|p^4]], [[Classification of groups of prime-fifth order|p^5]], [[Classification of groups of prime-sixth order|p^6]], [[Classification of groups of order two times a product of two distinct primes|2pq]], [[Classification of groups of order product of three distinct primes|pqr]] (all distinct) | General orders: [[Classification of groups of order a product of two distinct primes|pq]], [[Classification of groups of an order two times a prime|2p]], [[Classification of groups of order four times a prime|4p]] ([[Classification of groups of order four times a prime congruent to 1 modulo 4|p is congruent to 1 mod 4]], [[Classification of groups of order four times a prime congruent to 3 modulo 4|p is congruent to 3 mod 4]]), [[Classification of groups of prime-cube order|p^3]], [[Classification of groups of prime-fourth order|p^4]], [[Classification of groups of prime-fifth order|p^5]], [[Classification of groups of prime-sixth order|p^6]], [[Classification of groups of order two times a product of two distinct primes|2pq]], [[Classification of groups of order product of three distinct primes|pqr]] (all distinct) [[Classification of groups of order a product of a prime-square and another prime | p^2 q]] | ||
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Revision as of 21:16, 11 November 2023
I'm a third year maths student in the UK.
I intend to fill out this wiki with groups related stuff I know from my degree.
Representation theory articles
I'm contributing to articles on representation theory as I learn it - in particular I am trying to add examples to example-less articles on definitions, or applications of results.
Contributions
Non-exhaustive list.
- Completely reducible linear representation
- Representation of finite group over field of characteristic zero decomposes as direct sum of irreducible subrepresentations
- Irreducible complex representation of abelian group is one dimensional
- Kernel of representation of finite group is set of elements where the character evaluates to dimension of representation
- Representation is irreducible if and only if inner product of character is 1
- Character of a linear representation - much of the content on this page.
- Sum of irreducible representation on conjugacy class is scalar multiple of identity matrix
- Linear representation theory of general affine group:GA(1,5)
- Sum of elements in row of character table of finite group is non-negative integer
- Correspondence between normal subgroups and the kernels of characters of a finite group
- Number of real conjugacy classes is equal to number of irreducible real characters
Mackey theory
Other contributions
Non-exhaustive list.
- Group of units modulo n
- Group of units modulo prime power is cyclic (statement. Proof to be filled in)
- The special case Group of units modulo prime is cyclic of the above. Statement and proof.
- Group of units modulo two times prime power is cyclic (statement. Proof to be filled in)
- Cyclic group:Z21 and the majority of the content on frobenius group of order 21
- Automorphism group of Zp for p prime is isomorphic to Z(p-1) (statement, proof to be filled in)
- Derived subgroup of dihedral group
- Derived subgroup is trivial if and only if group is abelian
- Class function (links and vector spaces bit)
- Vector space
- Automorphism group of cyclic group
- Generalized quaternion group:Q16's Cayley table
- Orbit under group action
- Homomorphic image of subgroup is subgroup
- Self-inverse conjugacy class
- Galois group of a polynomial
- Primitive element theorem
- Galois field extension
- Groups with same character table need not be isomorphic
- Gnu function
- Abelian number
Groups of small orders
Further information: Table of number of groups for small orders
For my own persual if I need to look something up, but also may help you, pages for groups of small orders are in this table below. I intend to create more of these pages, where the order is sufficiently interesting (primes are not so interesting, neither are 2*primes, or primes^2, for example.)
General orders: pq, 2p, 4p (p is congruent to 1 mod 4, p is congruent to 3 mod 4), p^3, p^4, p^5, p^6, 2pq, pqr (all distinct) p^2 q
Also useful is the Classification of abelianness-forcing numbers