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

1. The group Z of all integers; the group law is the given by the addition 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 multiplication. We have homomorphisms exp : R → R× and exp : C → C× given by the exponential function. 5. The quotient group Q/Z is an inﬁnite abelian (or commutative) group in which each element has ﬁnite order. The next batch of examples are some matrix groups. In these examples we will ﬁx a coeﬃcient ring 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 commutative ring. 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 integer n. In the examples below the groups operate on the set 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 linear group GLn(R), consisting of all invertible n × n matrix with entries in the ring R. This set forms a group under matrix multiplication. 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 special linear group SLn(R) is the normal subgroup of GLn(R), consisting of all n × n matrics A with coeﬃcients in R such that det(A) = 1. It is the kernel of the × homomorphism 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 normal subgroup of Bn(R). When n = 3, Nn(R) is often called a “Heisenberg group”.

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 group action 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 orthogonal group.

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 complex conjugate 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 unitary group. The intersection Un ∩ SLn(C) is called the special unitary group. In the next batch of examples we shall ﬁnd some non-commutative ﬁnite groups.

12. The quaternion group Q; it has eight element:

Q = {±1, ±i, ±j, ±k} .

It is the group of all invertible elements in the ring of quaternions with integer coef- ﬁcents. The center 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 quaternion group 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 symmetric group of all permutations of the set {1, . . . , n} with n elements. (A permutation of a set T is a bijection 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 square roots of unity. The kernel of sign, consisting of all even permutations, is called the alternating group 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 dihedral group D2n is the group of all symmetries of Pn. It has 2n elements, consisting of n rotations 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 Platonic solid 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 tetrahedrons, the cubes, the regular octahedrons, the regular dodecahedrons and the regular icosahedrons. Then standard duality construction shows that the symmetry 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.

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