# Center of binary von Dyck group has order two

From Groupprops

## Statement

Define the group:

.

Then the element has order two if either of these hold:

- .

## Facts used

## Proof for

Follows from fact (2).

## Proof for the remaining cases

Much of the proof is common between the cases . Thus, with the exception of Steps (8)-(10), all other steps are generic to all .

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

1 | is in the center | commutes with all the generators since it is a power of each of them | |||

2 | -- | Cancel from both sides | |||

3 | [SHOW MORE] | ||||

4 | Step (3) | [SHOW MORE] | |||

5 | Let . | ||||

6 | . | Steps (4), (5) | |||

7 | Step (5) | [SHOW MORE] | |||

8 | If , then is conjugate to , and hence | Step (5) | [SHOW MORE] | ||

9 | If , then is conjugate to , and hence | Step (5) | [SHOW MORE] | ||

10 | If case, then is conjugate to , and hence | Step (5) | [SHOW MORE] | ||

11 | If , we have , where | Steps (6)-(10) | [SHOW MORE] | ||

12 | is isomorphic to a quotient of the dicyclic group with parameter , because it satisfies all the relations for that group, with . | Step (11) | |||

13 | Fact (2) | Follows from the previous step and Fact (2). | |||

14 | has order exactly two, i.e., it is not exactly the identity element | [SHOW MORE] |