S.P.E.A.K.

We are the Society for the Promotion of Education and Advancement of Knowledge.

# Abstract algebra

George

Posts : 222
Join date : 2017-11-26

Abstract algebra is my favourite area of mathematics. It is the study of abstract algebraic structures, most notably groups, rings and fields.

A group is a pair (G,·) consisting of a non-empty set G and a binary operation · on G such that · is associative in G, there is an identity element in G with respect to · (i.e. an element eG such that g·e = e·g = g for all gG) and every element in G has an inverse element in G with respect to · (i.e. for all gG there exists g−1G such that g·g−1 = g−1·g = e). (Note: the identity element of a group is unique, and given any element in the group its inverse is unique.) Often, when the binary operation is understood, the set G itself (rather than the pair (G,·)) is called a group. Note that while · has to be associative, it need not be commutative; if it is, then G is said to be an Abelian group under ·.

A ring is a triple (R,+,×) consisting of a non-empty set R and two binary operations on R called addition + and multiplication × such that (R,+) is an Abelian group, × is associative in R and distributive over + (i.e. for all a, b, cR, a×(b+c) = (a×b)+(a×c) and (b+ca = (b×a)+(c×a)). The additive identity is denoted 0R; many authors also insist that a ring must have a multiplicative identity, 1R. Again, if the operations + and × are clear from the context, the set R itself, rather than the cumbersome triple (R,+,×), is referred to as a ring.

A field is a ring (F,+,×) (as defined above) such that (F\{0F},×) is an Abelian group. And, once again, it is usual to call just F itself a field if + and × are understood.

In 2001, I celebrated my love for abstract algebra by composing an English sonnet about groups, rings and fields.

Lagrange’s theorem states for finite G
That every subgroup’s order must divide
G’s – if G’s prime then simple must G be.
All finite simple groups are classified.
Let R be any ring. All subrings I
Closed under multiplying are ideals.
One can construct then quotient rings R by
I – good for $\mathbb{Z}$ (a subset of the reals).
Real numbers form a field – and this consists
Of quite a structure one can marvel at.
Why, given each prime power, there exists
A Galois field of order simply that.
Nice is the study of groups, rings, and fields –
Great satisfaction to the mind it yields.

_________________
You are welcome to S.P.E.A.K.

Sophie

Posts : 31
Join date : 2017-12-02
Indeed: Group theory, ring theory, and field theory can be combined to produce more fascinating branches of abstract algebra – thus, Galois theory is a combination of field theory and group theory (particularly the theory of normal subgroups) and algebraic-number theory is a combination of field theory and ring theory (particularly the theory of ideals).

George

Posts : 222
Join date : 2017-11-26

And representation theory is a combination of group theory and linear algebra!

During my first year at Queen Mary and Westfield College – now Queen Mary, University of London – my course adviser was Professor Stephen Donkin. (He was actually only Doctor Donkin when my course started – but was very soon promoted to Professor Donkin.) And his area of specialism was representation theory. Indeed, the theory of Galois representations holds pride of place in the history of mathematics, being one of the tools employed by Andrew Wiles in his groundbreaking proof of Fermat’s last theorem – indirectly by proving the special case of the Taniyama–Shimura conjecture for semistable elliptic curves over $\mathbb{Q}$ (in particular the Frey elliptic curve).

_________________
You are welcome to S.P.E.A.K.

Sophie

Posts : 31
Join date : 2017-12-02
The Taniyama–Shimura conjecture is not a conjecture any more: since Wiles’s proof of the special case, it has been proved for the general case and is now known as the modularity theorem. What it asserts is that every elliptic curve over $\mathbb{Q}$ is modular.

An elliptic curve over $\mathbb{Q}$ is a curve with the equation y2 = f(x), where f(x) is a cubic polynomial in x with rational coefficients. And modular forms are related to transformations of the extended complex plane $\mathbb{C}\cup\{\infty\}$ of the form

$f(z)\,\,=\,\,\frac{az+b}{cz+d}$

where a, b, c, d are integers and adbc = 1. Not surprisingly, these transformations are related to the special linear group $\mathrm{SL}_2(\mathbb{Z})$ of 2×2 matrices with integer entries and unit determinant – indeed it turns out that the set of all these transformations form a group isomorphic to the projective special linear group $\mathrm{PSL}_2(\mathbb{Z})=\mathrm{SL}_2(\mathbb{Z})/\{I_2,-I_2\}$ where I2 is the identity 2×2 matrix. Now if, given a positive integer N, we define the subgroup Γ0(N) of $\mathrm{SL}_2(\mathbb{Z})$ as comprising those matrices such that N divides c, then Γ0(N) acts on the complex upper half-plane (complex numbers with non-negative imaginary part) – and a modular function of level N is a function on the complex upper half-plane that is invariant under Γ0(N) (and having a few other properties).

George

Posts : 222
Join date : 2017-11-26

Let us look more closely at the complex transformation

$f(z)\,\,=\,\,\frac{az+b}{cz+d}$

where the coefficients are integers with adbc = 1. There are two cases to consider: c = 0, and c ≠ 0.

Case 1: c = 0
In this case, due to the condition adbc = 1, we simply get the linear translations f(z) = z ± b for all z$\mathbb{C}\cup\{\infty\}$.

Case 2: c ≠ 0
Then f(z) is defined for all z$\mathbb{C}$ except z = −d/c. Also

$w\,\,=\,\,\frac{az+b}{cz+d}$

implies

$z\,\,=\,\,\frac{-dw+b}{cw-a}$

which is not defined for w = a/c. To get round these little obstacles, we simply set f(−d/c) = ∞ and f(∞) = a/c! Mathematicians need to be clever sometimes.

_________________
You are welcome to S.P.E.A.K.

Sophie

Posts : 31
Join date : 2017-12-02
And ∞, called the point at infinity, is not some inscrutable mathematical object plucked out of nowhere just to “make things work”: it can be constructed rigorously. Define a relation ~ on the set $\mathbb{C}\times\mathbb{C}\,\backslash\,\{(0,0)\}$ by (z,w)~(u,v) iff zv = uw. This is an equivalence relation, and the extended complex plane is the set of all equivalence classes under this relation. Denote the equivalence class containing (z,w) by [z,w]. The point at infinity ∞ is defined to be the equivalence class [z,0] (z ≠ 0) and $\mathbb{C}$ itself is identified with the set of equivalence classes $\{[z,1]:z\in\mathbb{C}\}$. Note that for all complex numbers z, w with w ≠ 0, [z,w] = [z/w,1]. Hence we can think of the extended complex plane as the extension of the complex plane by a point at infinity, $\mathbb{C}\cup\{\infty\}$.

We can define arithmetic in the extended complex plane by

$[z,w]+[u,v]=[zv+uw,wv]$ and $[z,w]\cdot[u,v]=[zu,wv]$

with subtraction and division being the inverse operations. One can check that these operations are well defined. Writing z for the equivalence class [z,1], we then have z ± ∞ = ∞ and z/∞ = 0 for all complex z, and z·∞ = ∞ and z/0 = ∞ for all nonzero complex z; also ∞·∞ = ∞, 0/∞ = 0, ∞/0 = ∞, but ∞ ± ∞, 0·∞, ∞/∞ and 0/0 are left undefined.

George

Posts : 222
Join date : 2017-11-26

The extension process has the unfortunate effect of removing certain algebraic properties from $\mathbb{C}$, especially the fact that the extended complex plane is no longer a field (∞ does not have an additive or multiplicative inverse). But the addition of a point at infinity has the advantage of enriching certain topological properties of the complex plane, especially the fact that the extended complex plane is compact (whereas $\mathbb{C}$ is not). Indeed $\mathbb{C}\cup\{\infty\}$ is homeomorphic to the surface of a sphere; it is also called the Riemann sphere.

But this is steering the thread away from the main topic, which is abstract algebra, to the realms of topology and projective geometry. Let us return to the main topic – though I would not mind delving into algebraic topology, particularly homotopy theory, myself.

_________________
You are welcome to S.P.E.A.K.

Sophie

Posts : 31
Join date : 2017-12-02
I have just read about Cayley, Sir George (1773–1857) in the glossary of essential information in Chapter 21, “Man the inventor”, of the section THE HUMAN ACHIEVEMENT in Volume 2, THE STUDY OF MAN, of the Reader’s Digest Library of Modern Knowledge. He was a “British landowner and inventor, who was the first man to identify the principles of heavier-than-air flight …, and to make a practical machine conforming to them”.

More relevantly for this thread, he was a distant uncle of the mathematician Arthur Cayley (1821–95). Permutation groups were studied before groups came to be investigated in abstraction, and it was Cayley who realized that every abstract group is isomorphic to some permutation group – a result that is now a theorem named after him.