Equivalence of presentations of dicyclic group
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
Consider the group with presentation:
- 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 .
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]|