, or, Groups to Play With Notes for Math 370 Ching-Li Chai First in the list are some commutative groups.

1. The Z of all ; the group law is the given by the of integers.

2. The group Z/nZ, where n ∈ N>0. It has n elements. The group law is the addition. If e1 em n = p1 ··· pm , where p1, . . . , pm are distinct primes and e1, . . . , em are positive integers, ∼ e1 em then Z/nZ = (Z/p1 Z) × · · · × (Z/pm Z). 3. The group Q (resp. R, resp. C) of all rational (resp. real, resp. complex) numbers under addition.

4. The group Q× (resp. R×, resp. C×) of all non-zero rational (resp. real, resp. complex) numbers under . We have exp : R → R× and exp : C → C× given by the exponential . 5. The group Q/Z is an infinite abelian (or commutative) group in which each has finite . The next batch of examples are some groups. In these examples we will fix a coefficient R. For now R is one of Q, R, C, Fp := Z/pZ or Z/nZ. But in fact one can take R to be any . Let R× be the group of all invertible element of R; so an element a ∈ R is in R× if and only if there exists an element b ∈ R such that a · b = 1 in R. We will also fix a positive n. In the examples below the groups operate on the Rn of all column vectors of size n with entries in R. So they operate as linear transformations on Rn. When R is infinite, these groups are usually infinite.

6. The general GLn(R), consisting of all invertible n × n matrix with entries in the ring R. This set forms a group under . Note that an n × n matrix A matrix with entries in R is invertible (i.e. it has an inverse matrix B in R) if and only if det(A) ∈ R×; the inverse A−1 of A is given by the standard formula in terms of the determant of A and its (n − 1) × (n − 1) minors.

7. The SLn(R) is the normal of GLn(R), consisting of all n × n matrics A with coefficients in R such that det(A) = 1. It is the of the × det : GLn(R) → R = GL1(R).

8. The subgroup Bn(R) of GLn(R), consisting of all upper-triangular n×n matrices with entries in R.

9. The group Nn(R) of Bn(R), consisting of all strictly upper-triangular n × n matrices with coefficients in R. The group Nn(R) is a of Bn(R). When n = 3, Nn(R) is often called a “”.

1 Here are some more matrix groups. They are examples of Lie groups, groups which can be studied using the tools of differential and integral calculus.

n 10. Denote by On(R) the orthogonal on R . It consists of all n × n matrices A with entries in R, such that

t t A · A = A · A = Idn .

The group law is induced by matrix multiplication. The intersection On(R) ∩ SLn(R) is called the special .

11. For any n × n matrix A with entries in C, denote by A∗ its hermitian conjugate, that is the result of first taking the of A (i.e. conjugate all entries of A), than take the transpose. Denote by Un the set of all n × n matrices A with entries in ∗ ∗ C, such that A · A = A · A = Idn. It is a subgroup of GLn(C), called the . The intersection Un ∩ SLn(C) is called the . In the next batch of examples we shall find some non-commutative finite groups.

12. The group Q; it has eight element:

Q = {±1, ±i, ±j, ±k} .

It is the group of all invertible elements in the ring of with integer coef- ficents. The of Q is {±1}. We have i2 = j2 = k2 = −1, i · j = k = −j · i, j · k = i = −k · j and k · i = j = −i · k. The Q is related to the Hamiltonian quaternions as follows. Let

HZ = {a + b · i + c · j + d · k | a, b, c, d ∈ Z} be the ring of Hamiltonian quaternions with integer coefficients. Then Q is the group H× of invertible elements in H.

13. Let n ≥ 1 be a positive integer. Denote by Sn the of all of the set {1, . . . , n} with n elements. (A of a set T is a from T to itself.) The group law for Sn is given by composition of permutations. In order to conform with the standard convention about permutations, we will think of Sn as operation on the right of {1, . . . , n}. So for σ1, σ2 ∈ Sn, their product σ1 · σ2 is the permutation which sends each element m ∈ {1, . . . , n} to ((m)σ1)σ2. In other words, permute first by σ1 and then by σ2. The group Sn has n! elements.

14. We can attach to every permutation σ ∈ Sn its sign. The sign of a transposition is −1, and sign : Sn → µ2 := {±1} is a surjective homomorphism from Sn to the group µ2 = {±1} of the roots of unity. The kernel of sign, consisting of all even permutations, is called the in n letters.

2 15. The group S4 contains a copy of the Klein-four group (Z/2Z) × (Z/2Z), namely the normal subgroup N = {id, (12)(34), (13)(24), (14)(23)}. Here (12)(23) is the permu- tation of the set {1, 2, 3, 4} which interchanges 1, 2 and interchanges 3, 4; similarly for the two other non-trivial elements. The quotient of S4 by this normal subgroup N is isomorphic to S3. Notice that N is also a subgroup of A4.

2 16. Let n ≥ 3 be a positive integer, and let Pn be the regular n − gon on R , with vertices 2πj 2πj  cos( n ), sin( n ) , j = 0, 1, . . . , n − 1. The D2n is the group of all of Pn. It has 2n elements, consisting of n and n reflections. It can be generated by two element s and t, such that s is an element of order 2 (a reflection), t is an element of order n (a reflection), and s · t · s−1 = t−1. When n = 3, the dihedral group D6 is isomorphic to the symmetric group S3. 17. More generally, let S be a in R3. Then the group of all symmetries of S is a finite subgroup of the orthogonal group O3(R). Here is a list of the Platonic solids: the regular , the , the regular , the regular dodecahedrons and the regular icosahedrons. Then standard construction shows that the groups for the cubes and for the regular octahedrons are isomorphic; also the symmetry groups for the regular dodecahedrons and the regular icosohedrons are isomorphic. Artin’s book is an excellent source for informations about these groups.