## Experimental Mathematics

- Experiment. Math.
- Volume 18, Issue 2 (2009), 201-204.

### On Symmetry of Flat Manifolds

#### Abstract

We give an example of a Bieberbach group $\Gamma$ for which $\Out(\Gamma)$ is a cyclic group of order $3$. We also calculate the outer automorphism group of a direct product of $n$ copies of a Bieberbach group with trivial center, for $n \in \mathbb{N}$. As a corollary we get that every symmetric group can be realized as an outer automorphism group of some Bieberbach group.

#### Article information

**Source**

Experiment. Math., Volume 18, Issue 2 (2009), 201-204.

**Dates**

First available in Project Euclid: 25 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1259158430

**Mathematical Reviews number (MathSciNet)**

MR2549689

**Zentralblatt MATH identifier**

1202.20052

**Subjects**

Primary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25]

Secondary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 57S30: Discontinuous groups of transformations

**Keywords**

Bieberbach groups directly indecomposable groups

#### Citation

Lutowski, Rafał. On Symmetry of Flat Manifolds. Experiment. Math. 18 (2009), no. 2, 201--204. https://projecteuclid.org/euclid.em/1259158430