**Definition**. Let be a commutative ring with 1. The** Heisenberg group** is defined by

Clearly is a group under matrix multiplication. Thus is a subgroup of the group of invertible upper triangular matrices with entries from

**Problem 1**. Let be a prime number and let the field of order Let be the group of invertible upper triangular matrices with entries from Prove that is the unique Sylow -subgroup of

**Solution**. Let Since is invertible we must have and so Thus, since the entries of come from there are possible values for and possible values for Thus It is obvious that Since we see that is indeed a Sylow -subgroup of To show that is the only Sylow -subgroup of we only need to show that is normal in because, by Sylow theorem, Sylow -subgroups are conjugates. To prove that is normal in suppose that is the multiplicative group of Let Define by

for all It is straightforward to check that is a group homomorphism and Thus is a normal subgroup of

**Notation**. Let be a commutative ring with 1. For the sake of simplicity, let’s use this notation

So

**Remark**. ** **Multiplying two elements of will gives us the following identities

**Problem 2**. Let be a commutative ring with 1 and suppose that and Prove that

1) and are subgroups of and is normal in

2) i.e. is the semidirect product of and

**Solution**. 1) Let be two elements of Then by the above remark

So is a subgroup of Similarly, is a subgroup of Now, let be any element of Then So is normal in Note that in general and so is not normal in

2) As we showed in the first part, is normal in and clearly and