# Checker for J4 # Check orders from definition chor 1 2 chor 2 4 mu 1 2 3 chor 3 37 mu 3 2 4 mu 3 4 5 chor 5 10 # Find a 2A element z by powering up an element of order 24. # Then az^g has odd order for some g, so a is in 2A as well. mu 3 5 6 chor 6 24 pwr 12 6 7 mu 4 2 8 mu 8 8 9 cj 7 9 10 mu 1 10 11 chor 11 11 # Construct some elements in C(b^2) (an involution centralizer) mu 3 8 12 mu 3 3 13 mu 12 13 14 mu 2 2 15 cj 15 14 16 mu 15 16 17 pwr 5 17 18 mu 14 18 19 # This element is in C(b^2) mu 4 8 20 mu 20 4 21 mu 21 3 22 cj 15 22 23 mu 15 23 24 mu 22 24 25 # This element is in C(b^2) # Use these elements to construct an element in C(b) with # order divisible by 5. The centralizers of 4B and 4C have # orders which are not divisible by 5, which proves that # b is in 4A. mu 25 19 26 mu 19 26 27 pwr 3 27 28 pwr 4 26 29 mu 28 29 30 mu 30 25 31 chor 31 20 # This element has order dividing 5... com 31 2 32 chor 32 1 # ...and it commutes with b.