We can determine exactly which positive numbers occur as the number of Sylow p-subgroups of finite groups, at least modulo some detailed information about simple groups. What can we say about the number of subgroups of order p?
Is Sylow’s theorem sharp?
Sylow’s theorem says the number of Sylow p-subgroups is congruent to 1 mod p. Does the converse hold? If n is a number congruent to 1 mod p, then is there a finite group with n Sylow p-subgroups?
Measuring current
How do batteries work? How can we run a circuit for a long time on a small battery?
Fusion examples
If two elements of an abelian Sylow P are conjugate in G, then they are conjugate in NG(P), but in GL(3,q) it can require conjugation in two separate normalizers. Is there a group that requires three?
Sylow numbers
A group has either 0, ā, or an odd number of involutions, and every such number occurs. A group either has 0, ā, or (pā1) mod 2p elements of order p, p odd. Does every such number occur?
Sylow intersections
Are halfway tame intersections automatically tame? Are maximal sylow intersections of maximum order?
Struggling to create good examples
How easy should examples be?
QR Codes in javascript
Can we generate QR codes directly in javascript?
Endomorphism rings of finite length modules
Endomorphisms of finite length modules just move the factors around, right?
Ī© covers of finite groups
Is every finite group the quotient of a finite group by the subgroup generated by elements of small order?