FinGrp.yaml

id: FinGrp
name: category of finite groups
notation: $\FinGrp$
objects: finite groups
morphisms: group homomorphisms
description: null
nlab_link: https://ncatlab.org/nlab/show/finite+group

tags:
  - algebra

related_categories:
  - FinAb
  - Grp
  - Grp_c

satisfied_properties:
  - property_id: locally small
    reason: It is a full subcategory of $\Grp$, which is locally small.

  - property_id: pointed
    reason: The trivial group is a zero object.
    check_redundancy: false

  - property_id: essentially countable
    reason: The underlying set of a finite structure can be chosen to be a subset of $\IN$.

  - property_id: mono-regular
    reason: See Prop. 4.2 at the <a href="https://ncatlab.org/nlab/show/regular+monomorphism#Examples" target="_blank">nLab</a>. The proof also works for finite groups.

  - property_id: coequalizers
    reason: The quotient group of a finite group is still finite.

  - property_id: Malcev
    reason: A direct argument is possible, but this can also be derived from the observation that $\FinGrp$ is the category of group objects in $(\FinSet,\times)$ and Example 2.2.16 in <a href="https://ncatlab.org/nlab/show/Malcev,+protomodular,+homological+and+semi-abelian+categories" target="_blank">Malcev, protomodular, homological and semi-abelian categories</a>.

  - property_id: regular
    reason: The category is Malcev and hence finitely complete, and it has all coequalizers. The regular epimorphisms coincide with the surjective group homomorphisms (see below), hence are clearly stable under pullbacks.

  - property_id: conormal
    reason: Since epimorphisms are surjective (see below), this is the first isomorphism theorem for finite groups.

  - property_id: ℵ₁-accessible
    reason: The proof works exactly as for <a href="/category/FinSet">$\FinSet$</a>.

  - property_id: effective congruences
    reason: 'Suppose we have a congruence $f, g : E \rightrightarrows X$ in $\FinGrp$. Since the embedding $\FinGrp \hookrightarrow \Grp$ preserves finite limits, it is also a congruence in $\Grp$. We already know that <a href="/category/Grp">$\Grp$</a> has effective congruences since it is algebraic. Using <a href="/lemma/effective-congruence-quotients">this result</a>, we see that $E$ is the kernel pair of $X \to (X/E)_{\Grp}$ in $\Grp$. Also, the quotient $(X/E)_{\Grp}$ is finite; and the forgetful functor $\FinGrp \to \Grp$ is fully faithful <a href="https://ncatlab.org/nlab/show/reflected+limit#FullSubcategoryInclusionReflectCoLimits" target="_blank">and therefore reflects limits</a>. Thus, we conclude that $E$ is the kernel pair of $X \to (X/E)_{\Grp}$ in $\FinGrp$ as well.'

unsatisfied_properties:
  - property_id: normal
    reason: Every non-normal subgroup of a finite group (such as $C_2 \hookrightarrow S_3$) provides a counterexample.

  - property_id: cogenerator
    reason: 'We apply <a href="/lemma/missing_cogenerator">this lemma</a> to the collection of finite simple groups: Any non-trivial homomorphism from a finite simple group to a finite group must be injective, and for every $n \in \IN$ there is a finite simple group of size $\geq n$ (for example, the alternating group on $n+5$ elements).'

  - property_id: skeletal
    reason: This is trivial.

  - property_id: binary copowers
    reason: 'Assume that $C_2 \sqcup C_2$ exists. This is a finite group, say of order $N$, with two involutions $u,v$ such that for every finite group $G$ with two involutions $a,b$ there is a unique homomorphism $\varphi : C_2 \sqcup C_2 \to G$ with $\varphi(u)=a$ and $\varphi(v)=2$. In particular, when $G$ is generated by $a,b$, then $\ord(G) \leq N$. But then the dihedral group $G := D_N$ of order $2N$ yields a contradiction.'

  - property_id: small
    reason: Even the collection of trivial groups is not small.

  - property_id: generator
    reason: If $A,B$ are finite groups whose orders are coprime, then we know that $\Hom(A,B)$ is trivial. But a generator would admit a non-trivial homomorphism to any other non-trivial finite group.

  - property_id: sequential colimits
    reason: This follows from <a href="/lemma/special_sequential_colimits">this lemma</a>.

  - property_id: countable
    reason: This is trivial.

special_objects:
  initial object:
    description: trivial group
  terminal object:
    description: trivial group
  products:
    description: '[finite case] direct products with pointwise operations'

special_morphisms:
  isomorphisms:
    description: bijective homomorphisms
    reason: This follows exactly as for <a href="/category/Grp">$\Grp$</a>.
  monomorphisms:
    description: injective homomorphisms
    reason: 'Let $f : A \to B$ be a monomorphism of finite groups. Let $a \in A$ be in the kernel of $a$, say of order $n$. Then we may view $a$ as a morphism $a : C_n \to A$ with $f \circ a = 1$ (the trivial homomorphism), and $C_n$ is finite. Hence, $a = 1$.'
  epimorphisms:
    description: surjective homomorphisms
    reason: 'For the non-trivial direction, if $f : G \to H$ is an epimorphism, we may factor it as $G \to f(G) \to H$, and $f(G) \to H$ is still an epimorphism, but also an inclusion and hence a monomorphism. Since we already know that the category is mono-regular, $f(G) \to H$ must be an isomorphism.'
  regular monomorphisms:
    description: same as monomorphisms
    reason: This is because the category is mono-regular.
  regular epimorphisms:
    description: same as epimorphisms
    reason: This is because the category is epi-regular.