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189-570A: Higher Algebra I
Assignment 1
Due: Monday, September 23.
1. Explain how the following assignments extend naturally to
functors by describing the induced maps $\hom(A,B) \rightarrow
\hom(F(A),F(B))$
on
$\hom$ sets.
State whether each functor is covariant or contravariant.
a. The assignment from the category ${\bf Grp}$
of groups to the category
${\bf Alg}_K$ of algebras over a field $K$, which to a group $G$
associates the group algebra
$K[G]$ consisting of the finite
$K$-linear combinations of elements of $G$, equipped with the
natural multiplication (extending by $K$-linearity the multiplication on $G$).
b. The assignment from the category
of finite groups to the category
of $K$-algebras which to a group $G$
associates the space of $K$-valued functions on $G$ equipped with the
convolution product.
c. The assignment from the category ${\bf VS}_K$ of vector spaces over $K$
to itself which to an object $V$ associates the dual space $V^*$ consisting
of the $K$-linear functionals on $V$.
2. Let ${\bf Mfld}$ be the category of real pointed differentiable
manifolds, whose objects consist of pairs $(M,x)$ where $M$ is a real differentiable manifold and $x$ is a point on $M$. The morphisms from $(M_1,x_1)$
to $(M_2,x_2)$ in this category are the differentiable maps from $M_1$ to $M_2$
sending $x_1$ to $x_2$.
Let $T$ be the assignment which to $(M,x)$ associates the tangent space of
$M$ at $x$. Show that $T$ extends to a covariant
functor from ${\bf Mfld}$ to
the category
${\bf VS}_{\bf R}$ of real vector spaces.
3. Let C be any category, and fix any object * of C.
a. Show that the assignment from C to Sets which to
$A \in {\rm Ob}({\bf C})$
associates the set $\hom(*,A)$
extends naturally to a covariant
functor from C to Sets.
b. Show that the assignment from C to Sets which to
$A\in {\rm Ob}({\bf C})$ associates the set $\hom(A,*)$
extends
naturally to a contravariant
functor from C to Sets.
4. Show that
a. If $0 \rightarrow V' \rightarrow V \rightarrow V'' \rightarrow 0$
is an exact sequence of vector
spaces, the resulting sequences
$$ 0 \rightarrow \hom(W,V') \rightarrow \hom(W,V) \rightarrow \hom(W,V'') \rightarrow 0, \qquad
0 \rightarrow \hom(V'',W) \rightarrow \hom(V,W) \rightarrow \hom(V',W) \rightarrow 0 $$
are also exact. (One says that the functors $\hom(W,-)$ and
$\hom(-,W)$ are exact.)
b. If VSK is replaced by the category
ModR of modules over a ring $R$,
show that the sequences
$$
0 \rightarrow \hom(W,V') \rightarrow \hom(W,V) \rightarrow \hom(W,V''),
\qquad
0 \rightarrow \hom(V'',W) \rightarrow \hom(V,W) \rightarrow \hom(V',W) $$
continue to be exact, but that
the $R$-module homomorphisms $ \hom(W,V)\rightarrow \hom(W,V'')$
and
$\hom(V,W) \rightarrow \hom(V',W)$
need not be surjective in general.
5. Let $p$ be a prime, and let $G$ be a finite group whose cardinality is not
a power of $p$. Suppose that $G$ is contained in a group
${\tilde G}$ which contains a Sylow $p$-subgroup $P$.
Show that $G$ has a proper subgroup of index prime to $p$.
(Hint: considering the left multiplication action of $G$ on the coset space
${\tilde G}/P$.)
6. Let $p$ be a prime and let ${\rm GL}_n(p)$ be the group of
invertible $n\times n$ matrices with entries in the field with $p$ elements.
Calculate the cardinality of this group, and
describe an explicit $p$-Sylow subgroup of it. How many
distinct Sylow $p$-subgroups does it possess?
7. Show that every finite group $G$ can be realised as a subgroup of
${\rm GL}_n(p)$ for a suitable $n$. Use the results of questions $5$ and
$6$ to conclude that $G$ contains a proper subgroup of index prime to $p$.
Use this to give an alternate proof of the existence part of Sylow's theorem
shown in class.
8. Show that the following three statements are equivalent:
(a) The transitive $G$-sets $X$ and $Y$ are isomorphic, as $G$-sets.
(b) The stabiliser subgroups ${\rm Stab}_G(x)$ and ${\rm Stab}_G(y)$
are conjugate in $G$, for all $x\in X$ and $y\in Y$.
(c) There exists $x\in X$ and $y\in Y$ for which ${\rm Stab}_G(x)$ is conjugate
to ${\rm Stab}_G(y)$ in $G$.
9. Let $K$ be a field of characteristic zero,
let $G$ be a finite group acting on a finite set $X$, and let
$V:=K[X]$ be the vector space of formal $K$-linear combinations of elements
of $X$, endowed with the
induced linear action of $G$.
a. Show that $\dim_{K}(V^G)$ is the number of orbits of $G$ acting on $X$.
b. For any $g\in G$, let ${\rm FP}(g)$ denote the
number of fixed points for $g$ acting on $X$. Show that the average
number of ${\rm FP}(g)$ as $g$ ranges over $G$ is equal to the number of
distinct $G$-orbits in $X$:
$$\frac{1}{\# G}\sum_{g\in G} {\rm FP}(g) = \#{\rm Orbits \ in\ } X.$$
c. Show that
the action of $G$ on $X$ is doubly
transitive, i.e., $G$
acts transitively on the complement of the
diagonal in the Cartesian product of
$X$ with itself,
if and only if
$$\frac{1}{\# G} \sum_{g\in G} {\rm FP}(g)^2 = 2.$$
10.
Let $X_1$ and $X_2$ be two transitive $G$-sets (for $G$ a finite group), and
suppose that $g$ has the same number of fixed points acting on $X_1$ and
on $X_2$, for all $g\in G$.
a. If $K$ is a field whose characteristic does not divide $\#G$,
show that the linear representations $K[X_1]$ and $K[X_2]$ are isomorphic.
(With notations as in question 9.)
b. Are the sets $X_1$ and $X_2$ necessarily isomorphic as $G$-sets? Prove this,
or give a counterexample if it is false in general.