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 N

_{G}(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?