We first present the isomorphism theorems of the
groups.
Theorem A (groups) Let G and H be groups, and let f : G \rightarrow H be a
homomorphism. Then: • The
kernel of f is a
normal subgroup of G, • The
image of f is a
subgroup of H, and • The image of f is
isomorphic to the
quotient group G / \ker f. In particular, if f is
surjective then H is isomorphic to G / \ker f. This theorem is usually called the
first isomorphism theorem.
Theorem B (groups) Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then the following hold: • The
product SN is a subgroup of G, • The subgroup N is a normal subgroup of SN, • The
intersection S \cap N is a normal subgroup of S, and • The quotient groups (SN)/N and S/(S\cap N) are isomorphic. Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the
normalizer of N in G. In this case, N is not a normal subgroup of G, but N is still a normal subgroup of the product SN. This theorem is sometimes called the
second isomorphism theorem, or the
parallelogram theorem. An application of the second isomorphism theorem identifies
projective linear groups: for example, the group on the
complex projective line starts with setting G = \operatorname{GL}_2(\mathbb{C}), the group of
invertible 2 × 2
complex matrices, S = \operatorname{SL}_2(\mathbb{C}), the subgroup of
determinant 1 matrices, and N the normal subgroup of scalar matrices \mathbb{C}^{\times}\!I = \left\{\left( \begin{smallmatrix} a & 0 \\ 0 & a \end{smallmatrix} \right) : a \in \mathbb{C}^{\times} \right\}, we have S \cap N = \{\pm I\}, where I is the
identity matrix, and SN = \operatorname{GL}_2(\mathbb{C}). Then the second isomorphism theorem states that: : \operatorname{PGL}_2(\mathbb{C}) := \operatorname{GL}_2 \left(\mathbb{C})/(\mathbb{C}^{\times}\!I\right) \cong \operatorname{SL}_2(\mathbb{C})/\{\pm I\} =: \operatorname{PSL}_2(\mathbb{C})
Theorem C (groups) Let G be a group, and N a normal subgroup of G. Then • If K is a subgroup of G such that N \subseteq K \subseteq G, then G/N has a subgroup isomorphic to K/N. • Every subgroup of G/N is of the form K/N for some subgroup K of G such that N \subseteq K \subseteq G. • If K is a normal subgroup of G such that N \subseteq K \subseteq G, then G/N has a normal subgroup isomorphic to K/N. • Every normal subgroup of G/N is of the form K/N for some normal subgroup K of G such that N \subseteq K \subseteq G. • If K is a normal subgroup of G such that N \subseteq K \subseteq G, then the quotient group (G/N)/(K/N) is isomorphic to G/K. The last statement is sometimes referred to as the
third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the
lattice theorem,
correspondence theorem, or
fourth isomorphism theorem.
Theorem D (groups) Let G be a group, and N a normal subgroup of G. The canonical projection homomorphism G\rightarrow G/N defines a bijective correspondence between the set of subgroups of G containing N and the set of (all) subgroups of G/N. Under this correspondence normal subgroups correspond to normal subgroups. This theorem is sometimes called the
correspondence theorem, the
lattice theorem, and the
fourth isomorphism theorem. The
Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.
Discussion The first isomorphism theorem can be expressed in
category theoretical language by saying that the
category of groups is (normal epi, mono)-factorizable; in other words, the
normal epimorphisms and the
monomorphisms form a
factorization system for the
category. This is captured in the
commutative diagram in the margin, which shows the
objects and
morphisms whose existence can be deduced from the morphism f : G \rightarrow H. The diagram shows that every morphism in the category of groups has a
kernel in the category theoretical sense; the arbitrary morphism
f factors into \iota \circ \pi, where
ι is a monomorphism and
π is an epimorphism (in a
conormal category, all epimorphisms are normal). This is represented in the diagram by an object \ker f and a monomorphism \kappa: \ker f \rightarrow G (kernels are always monomorphisms), which complete the
short exact sequence running from the lower left to the upper right of the diagram. The use of the
exact sequence convention saves us from having to draw the
zero morphisms from \ker f to H and G / \ker f. If the sequence is right split (i.e., there is a morphism
σ that maps G / \operatorname{ker} f to a -preimage of itself), then
G is the
semidirect product of the normal subgroup \operatorname{im} \kappa and the subgroup \operatorname{im} \sigma. If it is left split (i.e., there exists some \rho: G \rightarrow \operatorname{ker} f such that \rho \circ \kappa = \operatorname{id}_{\text{ker} f}), then it must also be right split, and \operatorname{im} \kappa \times \operatorname{im} \sigma is a
direct product decomposition of
G. In general, the existence of a right split does not imply the existence of a left split; but in an
abelian category (such as
that of abelian groups), left splits and right splits are equivalent by the
splitting lemma, and a right split is sufficient to produce a
direct sum decomposition \operatorname{im} \kappa \oplus \operatorname{im} \sigma. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence 0 \rightarrow G / \operatorname{ker} f \rightarrow H \rightarrow \operatorname{coker} f \rightarrow 0. In the second isomorphism theorem, the product
SN is the
join of
S and
N in the
lattice of subgroups of
G, while the intersection
S ∩
N is the
meet. The third isomorphism theorem is generalized by the
nine lemma to
abelian categories and more general maps between objects.
Note on numbers and names Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules. It is less common to include the Theorem D, usually known as the
lattice theorem or the
correspondence theorem, as one of isomorphism theorems, but when included, it is the last one. == Rings ==