MathDB

5

Part of 2008 IMC

Problems(2)

IMC 2008 Day 1 P5 - Automorphisms of Subgroups

Source: Problem 5

7/30/2008
Does there exist a finite group G G with a normal subgroup H H such that ∣Aut H∣>∣Aut G∣ |\text{Aut } H| > |\text{Aut } G|? Disprove or provide an example. Here the notation ∣Aut X∣ |\text{Aut } X| for some group X X denotes the number of isomorphisms from X X to itself.
group theoryabstract algebraIMCcollege contests
IMC 2008 Day 2 P5 - Determinant

Source: Problem 5

7/28/2008
Let n n be a positive integer, and consider the matrix A \equal{} (a_{ij})_{1\leq i,j\leq n} where a_{ij} \equal{} 1 if i\plus{}j is prime and a_{ij} \equal{} 0 otherwise. Prove that |\det A| \equal{} k^2 for some integer k k.
linear algebramatrixIMCcollege contests