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 | ![]() |
![]() |
Since every element commutes with its powers, ![]() ![]() ![]() ![]() ![]() | ||
2 | ![]() ![]() |
![]() |
We cancel a ![]() | ||
3 | ![]() |
![]() |
Step (2) | Combining ![]() ![]() ![]() | |
4 | ![]() |
Step (3) | Multiply both sides of Step (3) by ![]() ![]() | ||
5 | ![]() ![]() |
Fact (1) | ![]() |
Step (4) | [SHOW MORE] |
6 | ![]() ![]() |
![]() |
[SHOW MORE] | ||
7 | ![]() ![]() |
Steps (6),(7) | |||
8 | Setting ![]() ![]() ![]() |
![]() |
Steps (4), (7) | [SHOW MORE] | |
9 | Conversely, the original relations can be deduced from ![]() ![]() ![]() |
[SHOW MORE] |