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:
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.
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 | |
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1 | ![]() |
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2 | ![]() |
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-- | Cancel ![]() | |
3 | ![]() |
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4 | ![]() |
Step (3) | [SHOW MORE] | ||
5 | Let ![]() |
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6 | ![]() |
Steps (4), (5) | ![]() | ||
7 | ![]() |
Step (5) | [SHOW MORE] | ||
8 | If ![]() ![]() ![]() ![]() |
Step (5) | [SHOW MORE] | ||
9 | If ![]() ![]() ![]() ![]() |
Step (5) | [SHOW MORE] | ||
10 | If ![]() ![]() ![]() ![]() |
Step (5) | [SHOW MORE] | ||
11 | If ![]() ![]() ![]() |
Steps (6)-(10) | [SHOW MORE] | ||
12 | ![]() ![]() ![]() |
Step (11) | |||
13 | ![]() |
Fact (2) | Follows from the previous step and Fact (2). | ||
14 | ![]() |
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