# Equivalence of presentations of dicyclic group

From Groupprops

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

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

## Statement

Consider the group with presentation:

Then:

- The element is in the center of .
- is the identity element of .
- If we set , and denote by the identity element, satisfies the relations . Further, the presentation:

also defines .

## Facts used

## Proof

We use the notation as in the statement above.

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

1 | is central | Since every element commutes with its powers, all commute with . Since the group is generated by , the centralizer of is the whole group, so is central. | |||

2 | , so the group is generated by | We cancel a from both sides. | |||

3 | Step (2) | Combining and Step (2), we get , so . | |||

4 | Step (3) | Multiply both sides of Step (3) by on the left and on the right, then interchange the two sides. | |||

5 | , i.e., | Fact (1) | Step (4) | [SHOW MORE] | |

6 | , i.e., | [SHOW MORE] | |||

7 | so is the identity | Steps (6),(7) | |||

8 | Setting , , we get | Steps (4), (7) | [SHOW MORE] | ||

9 | Conversely, the original relations can be deduced from by setting , , so the two presentations define the same group | [SHOW MORE] |