Group, DYK Fact #100

[algebraic topology]

Did you know?

A set G is said to form a group under the binary operation ∘ iff:

1. ∘ is associative in G.
2. ∃ e ∊ G such that ∀ g ∊ G, g∘e = e∘g = g.
3. ∀ g ∊ G, ∃ g⁻¹ ∊ G such that g∘g⁻¹ = g⁻¹∘g = e.

Notes:

• A further axiom is sometimes stated: ∀ g₁, g₂ ∊ G, g₁∘g₂ ∊ G. This “closure property”, however, is already implicit in the definition of a binary operation. (A binary operation on a set S is defined as a function from S×S to S.)
• It follows that the element e is unique: it is called the identity element of G under ∘. Similarly, each g ∊ G has a unique g⁻¹, called the inverse element of g in G under ∘
• ∘ does not have to be commutative in G. If it is, G under the binary operation is said to be an Abelian group.
• For convenience, the symbol for the binary operation in a group is often left unwritten if it is clear which binary operation the group is associated with – thus g₁∘g₂ in G is simply written g₁g₂.

[JaneFairfax]

Did you know? The first mathematician to use the term “group” as it is used in mathematics today was Évariste Galois! At about that same time, Augustin Louis Cauchy was independently studying these structures, but did not use the same terminology as Galois. This is information I gather from the book A Course in Group Theory by John F. Humphreys.

[algebraic topology]

Did you know?

The first groups to be studied were permutation groups.

[JaneFairfax]

Did you know? A permutation on is a bijection from that set to itself. Under composition of mappings, the set of all these permutations forms a group of order n!, called the symmetric group of degree n, S_n.

Did you also know? Permutations can be odd or even. A permutation is odd if there are an odd number of ordered pairs in the set

and is even if the set above contains an even number of elements. Using this definition, it is easy to determine that the identity permutation is even and any transposition (permutation interchanging two distinct elements in and leaving everything else fixed) is odd. While this definition of odd and even permutations is intuitively easy to grasp, it can unfortunately be mathematically unwieldy. A much better definition, more elegant in an abstract sense and more useful in a practical way, is given in the book by Humphreys – which contains an excellent treatment of the subject of permutations.

[algebraic topology]

Did you know?

A k-cycle is even if and only if k is odd. This a permutation for which there exist k distinct integers such that for , and for any other i.

[JaneFairfax]

Did you know? If k is prime, every group of order k is cyclic. However, the converse is false: if every group of order k is cyclic, k is not necessarily a prime. For example, although 15 is not a prime, there are no noncyclic groups of order 15. This can be proved using the Sylow theorems.

For if G has order 15, the number of its Sylow 5-subgroups must divide 15 and be congruent to 1 modulo 5. This means that G has a unique Sylow 5-subgroup. Similarly G must have a unique Sylow 3-subgroup. Thus G has 1 element of order 1, 2 elements of order 3, and 4 elements of order 5. The other 8 elements must have order 15, and are therefore generators of the group G.

[algebraic topology]

Did you know?

Although the pioneering group theorists only studied permutation groups, the results they developed are equally applicable to abstract groups – because of Cayley’s theorem: Every group is isomorphic to a subgroup of a symmetric group.

[JaneFairfax]

Did you know? If H is subgroup of a group G, we can say that H is a maximal normal subgroup of G iff H is a proper subgroup of G and is normal in G, such that for any subgroup K, if H ◁ K and K ◁ G, then either H = K or K = G. A composition series for G is a finite chain of subgroups G₀, G₁, …, G_r with G₀ = G and G_r = <1_G> such that G_i is a maximal normal subgroup of G_i−1 for each i = 1, …, r.

[algebraic topology]

Did you know?

H is a maximal normal subgroup of G if and only if the factor group G/H is nontrivial and simple. A simple group is a group having no normal subgroups other than itself and the trivial subgroup. Hence the composition factors of a composition series are nontrivial and simple.

[JaneFairfax]

Did you know? If G₀ ▷ G₁ ▷ … ▷ G_r and H₀ ▷ H₁ ▷ … ▷ H_s are composition series of a group G, then r = s and there exists a bijection such that is isomorphic to for each i. This result is known as the Jordan–Hölder theorem.

[algebraic topology]

Did you know?

It is important to note that the Jordan–Hölder theorem does not say that every group has a composition series – only that if it has a composition series, then that composition series is unique up to isomorphism of the composition factors. Indeed, there are groups with no composition series at all, an example being the infinite cyclic group.

[JaneFairfax]

Did you know? The theorem was first discovered by the French mathematician Camille Jordan in 1870 and proved rigorously by the German mathematician Otto Hölder in 1889. Nowadays, however, the theorem can be easily proved using Schreier’s refinement theorem, which in turn can be easily proved using a result called the Zassenhaus lemma. This is how the Jordan–Hölder theorem is tackled in Chapter 15 of A Course in Number Theory by John F. Humphreys.

[algebraic topology]

Did you know?

Hans Zassenhaus was a German mathematician while Otto Schreier was an Austrian mathematician.

[JaneFairfax]

Did you know? If a group G has subgroups G = G₀, G₁, …, G_r = <1_G> such that G_i is a normal subgroup of G_i−1 and G_i−1/G_i is Abelian for all 1 ≤ i ≤ r, then G is said to be a soluble group.

[algebraic topology]

Did you know?

Americans call such a group “solvable” rather than “soluble”.

[JaneFairfax]

Did you know? If G is a soluble group, then any subgroup H of G is soluble, and for any normal subgroup N of G, G/N is soluble.

[algebraic topology]

Did you know?

If N is a normal subgroup of G such that N and G/N are both solvable, then G is solvable.

[JaneFairfax]

Did you know? Another way of looking at soluble groups G is to consider commutators in G. These are elements of the form xyx⁻¹y⁻¹ where x, y ∊ G. The derived group G′ of G is the subgroup generated by all the commutators in G. Now if we set G⁽⁰⁾ = G and G⁽¹⁾ = G′, the nth derived group of G is defined inductively by

G⁽ⁿ⁾ = G⁽ⁿ⁻¹⁾′ = <xyx⁻¹y⁻¹: x, y ∊ G⁽ⁿ⁻¹⁾>

Then G is soluble if and only if G^(r) = <1_G> for some positive integer r.

[algebraic topology]

Did you know?

Every nilpotent group is solvable. A nilpotent group is a group with a terminating upper central series.

[JaneFairfax]

Did you know? A finite group is nilpotent if and only if all its Sylow subgroups are normal and it is the internal direct product of all its Sylow subgroups.

[algebraic topology]

Did you know?

[JaneFairfax]

Did you know? There is another interesting class of groups between the nilpotent and soluble ones: the supersoluble groups. A supersoluble group is like a soluble group, except that the factor groups in its normal series are cyclic rather than Abelian. All supersoluble groups are soluble, and all finitely generated nilpotent groups are supersoluble.

Supersoluble groups are not mentioned in John F. Humphreys’s book A Course in Group Theory (at least not in what I have read of it so far) but I came across them today while browsing at Foyles in Central London in a book on finite groups written by a former lecturer of mine: Prof B.A.F. Wehrfritz of Queen Mary, University of London!

[algebraic topology]

Did you know?

All supersolvable groups are polycyclic, and all polycylic groups are solvable. A polycyclic group is like a supersolvable group, except that it only needs to have a subnormal rather than normal series with cyclic factors.

[JaneFairfax]

Did you know?

[algebraic topology]

Did you know?

The Feit–Thompson theorem states that all finite groups of odd order are solvable. This implies that every finite group of odd non-prime order is not simple.