# Periodicity is extension-closed

This article gives the statement, and possibly proof, of a group property (i.e., periodic group) satisfying a group metaproperty (i.e., extension-closed group property)

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## Statement

Suppose is a group and is a normal subgroup of such that both and the quotient group are periodic groups. Then, is also a periodic group.

Further, if is a set of primes such that both and are -groups -- all prime factors of orders of elements in these groups are in -- then is also a -group.

## Related facts

## Proof

We prove the second version. Note that taking to be the set of all primes will give the first version.

**Given**: A group . A normal subgroup with quotient group . A prime set such that both and are -groups: all elements of and all elements of have finite orders with all prime divisors of the order of every element in .

**To prove**: Every element of has finite order and every prime divisor of the order of every element is in .

**Proof**: We pick an arbitrary element . We let be the quotient map.

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

1 | has order equal to some -number as an element of . | is a -group. | |||

2 | where is the -number chosen in Step (1). | Step (1) | |||

3 | Let from Step (2). Then, has order equal to some -number as an element of . | is a -group | Step (2) | ||

4 | is the identity element of . Thus, the order of in is finite and divides the -number , hence it must also be a -number. | Step (3) |