# Equivalence of definitions of profinite group

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term profinite group

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equivalent for a topological group:

- It is the inverse limit of an inverse system of finite groups, each equipped with the discrete topology.
- It is a compact totally disconnected T0 topological group.

A topological group satisfying both equivalent conditions is termed a profinite group.

## Facts used

- Hausdorffness is product-closed
- Hausdorffness is hereditary
- Tychonoff's theorem
- Closed subset of compact space implies compact
- Connectedness is continuous image-closed
- Compact and totally disconnected implies every open neighborhood of identity contains an open normal subgroup
- Compact implies every open subgroup has finite index
- Compact to Hausdorff implies closed

## Proof

### (1) implies (2)

**Given**: An inverse system of finite groups with discrete topologies. is the inverse limit of the system.

**To prove**: is compact, , and totally disconnected.

**Proof**

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | can be identified with a closed subgroup of the product , equipped with the product topology and the direct product group structure. | definition of inverse limit | is an inverse limit of the s | ||

2 | is Hausdorff (hence ) | Facts (1), (2) | all the s are finite and discrete, hence Hausdorff | Step (1) | By Fact (1), the product is Hausdorff, hence by Fact (2), so is any subgroup of that. By Step (1), we thus get that is Hausdorff. |

3 | is compact | Facts (3), (4) | all the s are finite, hence compact | Step (1) | By Fact (3), the product is compact, hence by Fact (4), so is any closed subgroup of that. By Step (1), we thus get that is compact. |

4 | is totally disconnected | Fact (5) | all the s are discrete, hence totally disconnected | Step (1) | [SHOW MORE] |

### (2) implies (1)

**Given**: A compact totally disconnected group .

**To prove**: is the inverse limit of an inverse system of finite groups.

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The open normal subgroups of form a basis at the identity element for . | Fact (6) | is compact and totally disconnected | Follows directly from Fact (6). | |

2 | If is an open normal subgroup of , then is a finite group with the discrete topology. | Fact (7) | is compact | By Fact (7), is finite. Further, is open, and so are all its cosets, so the coset space gets a discrete quotient topology. | |

3 | Consider the homomorphism from to the inverse limit of the inverse system of quotients of by open normal subgroups. This is continuous and has dense image. | ||||

4 | The homomorphism of Step (3) is a closed map. | Fact (8) | |||

5 | The homomorphism of Step (3) is injective. | ||||

6 | The homomorphism of Step (3) is an isomorphism of topological groups. |