Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 7 October 6, 2005

Problem 1: Show that there are at most two non-isomorphic groups G of order pq, where p and q are primes with p < q. (Hint: G is either a direct or a semidirect product of its Sylow subgroups.)

Problem 2: Let G and F be groups and φ : G ⟶ F be a surjective group homomorphism. Let N = ker φ be the kernel of φ. Prove that G is a semidirect product with normal subgroup N if and only if there is a homomorphism s : F ⟶ G sich that φ ∘ s = id_{F}, the identity map on F. One says in this case that the short exact sequence

{e} ⟶ N ⟶ G ⟶^{φ} F ⟶ {e}

splits and s is called a split of φ.

Problem 3: Decide which of the following sets are subrings of the ring of all functions from the closed interval [0, 1] to C:

(a) The functions f(x) such that f(q) = 0 for all q ∈ [0, 1] ∩ Q.

(b) The polynomial functions.

(c) The functions which have a finite number of zeros together with the zero function.

(d) The functions wich have infinitely many of zeros.

(e) All functions f such that lim_{x→1^{−}} f(x) = 0.

(f) All rational linear combinations of the functions sin nx and cos nx, where n ∈ Z, n ≥ 0.

(g) All functions which are equal to a convergent power series expanded in 0 with radius of convergence larger than 1.

(h) The set of continous nowhere differentiable functions together with the differentiable functions.

Problem 4: (a) Show that the four dimensional real vector space R^{4} with base e, i, j and k becomes a noncommutative ring if we use the vector space addition as ring addition and if we indroduce a R-linear ring multiplication by requiring that e is a 1 and

i^{2} = j^{2} = k^{2} = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.

(b) Show that in the above ring every element besides 0 has a multiplicative inverse.

Problem 5^{∗}: Let n be a natural number n and p be a prime. Write

n = a_{0} + a_{1} p + a_{2} p^{2} + ⋯ + a_{k} p^{k}

with integers a_{i}, 0 ≤ a_{i} < p (this is the representation of n using the base p). Consider the regular action of Z_{p} = Z/pZ on itself by left multiplication. Using this action, define Z_{p}^{≀r}, r ≥ 1, inductively by Z_{p}^{≀1} = Z_{p}, Z_{p}^{≀k+1} = Z_{p}^{≀k} ≀Z_{p}. Prove that any p-Sylow subgroup of S_{n}, the symmetric group of degree n, is isomorphic to

\underbrace{Z_{p}^{≀1} × ⋯ × Z_{p}^{≀1}}_{a_{1}−times} × \underbrace{Z_{p}^{≀2} × ⋯ × Z_{p}^{≀2}}_{a_{2}−times} × ⋯ × \underbrace{Z_{p}^{≀k} × ⋯ × Z_{p}^{≀k}}_{a_{k}−times} .

(Hint: The exponent of p in the prime factor decomposition of n! is

[\frac{n}{p}] + [\frac{n}{p^{2}} ] + [\frac{n}{p^{3}} ] + ⋯,

where [x] denotes the largest integer ≤ x.)