# Normal of finite index implies completely divisibility-closed

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup of finite index) must also satisfy the second subgroup property (i.e., completely divisibility-closed normal subgroup)

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## Contents

## Statement

Suppose is a group and is a normal subgroup of finite index in , i.e., is a normal subgroup of and the index of in is finite. Then, is a completely divisibility-closed subgroup, and hence, a completely divisibility-closed normal subgroup of .

In other words, if is divisible by a prime number , then the quotient group is -torsion-free.

## Related facts

## Facts used

- Divisibility is inherited by quotient groups
- Finite and p-divisible implies p-powered (and hence also implies -torsion-free).

## Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

**Given**: A group , a normal subgroup of such that the index of in is finite (in other words, the quotient group is a finite group). is divisible by a prime .

**To prove**: is -torsion-free.

**Proof**:

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

1 | is -divisible. | Fact (1) | is normal in , is -divisible. | Given-fact combination direct. | |

2 | is -torsion-free. | Fact (2) | is normal of finite index, so is a finite group. | Step (1) | Step-fact-given direct. |