Abstract Algebra (Paper I, MAT CC 01,
Tilka Manjhi Univ.)
Unit 1: Groups – Homomorphisms, Actions, Series, and
Special Classes
Group Homomorphisms
A group homomorphism φ: G→H is a map between groups that preserves
the group operation. Equivalently, φ satisfies φ(e_G)=e_H and
φ(g₁g₂)=φ(g₁)φ(g₂) for all g₁,g₂∈G[1]. The kernel Ker(φ)={g∈G: φ(g)=e_H}
is a normal subgroup of G, and the image Im(φ)=φ(G) is a subgroup of H.
Key results include the First Isomorphism Theorem: G/Ker(φ)≅Im(φ) (so φ
induces an isomorphism of quotient and image). For example, the
determinant map det: GL(n,ℝ)→ℝ^* is a homomorphism with kernel SL(n,ℝ)
(matrices of determinant 1). Proofs of the isomorphism theorem use the
universal property of quotients: the kernel being normal ensures well-
definedness of the quotient map[2].
Group Actions
A group action of G on a set X is a map G × X → X , denoted ( g , x ) ↦ g ⋅ x ,
satisfying e ⋅ x =x and g ⋅ ( h ⋅ x )=( g h ) ⋅ x. Equivalently, each g∈G gives a
permutation of X and the map g↦(x↦g·x) is a homomorphism G→Sym(X).
The orbit of x is Orb_G(x)={g·x: g∈G} and the stabilizer is
Stab_G(x)={g∈G: g·x=x}. The Orbit–Stabilizer Theorem states that for finite
G,
$$|G| = |{\rm Orb}_G(x)|\,|{\rm Stab}_G(x)|,$$
so the orbit size divides |G|[3]. For example, a group acting on itself by
conjugation (g·x=gxg^{-1}) has orbits the conjugacy classes, and
Stab(x)=C_G(x), the centralizer. Group actions link to counting arguments
(e.g. counting colorings or symmetries) via Burnside’s Lemma.
Sylow Theorems
Let |G|=p^e m with p∤m. A Sylow p-subgroup is a subgroup of order p^e.
The three Sylow Theorems assert:
- Sylow I: G has at least one Sylow p-subgroup.
- Sylow II: All Sylow p-subgroups are conjugate in G.
- Sylow III: The number n p of Sylow p-subgroups satisfies n p ≡1(mod p) and
divides m.
, For instance, a group of order 21 (3·7) must have a unique Sylow-7 subgroup
(so it is normal) and either 1 or 3 Sylow-3 subgroups (each of order 3)[4]. A
typical proof (via group actions on left cosets) shows existence by counting
fixed points, and uniqueness up to conjugacy by action on the set of Sylow
subgroups. These theorems constrain possible group structures: e.g. if
n_p=1 then the Sylow p-subgroup is normal.
Normal and Subnormal Series; Composition Series
A normal series of G is a chain of subgroups
{1 }=N 0 ⊲ N 1 ⊲ ⋯ ⊲ N k =G ,
where each N i ⊲N i+1. A subnormal series only requires each N i normal in G (or
each normal in the next). A series is refinement of another if the chain is
strictly longer. A composition series is a subnormal series with each factor
N i+ 1 /N i simple (having no nontrivial normal subgroups)[5]. By a standard
argument using Zorn’s Lemma (or simply induction on |G|), every finite
group has a composition series.
Jordan–Hölder Theorem: Any two composition series of a finite group G
have isomorphic factor groups up to order. In particular, the list of simple
factors (Jordan–Hölder factors) is uniquely determined (up to permutation
and isomorphism)[6]. For example, S3 has a composition series 1 ⊲C 2 ⊲S 3 with
factors C 2 and C 3. A different series is not possible (since one of 2 or 3 must
appear). The JH theorem is proved by refining and comparing series using
Schreier’s theorem.
Solvable and Nilpotent Groups
A group G is solvable if it has a subnormal series whose factor groups are all
abelian[7]. Equivalently, its derived series G(0 )=G , G(1)= [ G , G ] , G(n +1)= [ G (n) , G(n )]
eventually reaches the trivial subgroup[7]. For finite G, this is also equivalent
to having a composition series whose factors are cyclic of prime order[8].
Examples include all abelian and all p-groups. Notably, S3 (of order 6) is
solvable, while A5 (of order 60) is not, since A5 has nonabelian composition
factors. The commutator subgroup [G,G] (generated by all commutators
−1 −1
x y x y ) measures how nonabelian G is. G/[G,G] is the maximal abelian
quotient; iterating commutators gives the derived series mentioned above.
Solvability of G⇔G/[G,G] abelian, etc.
A group G is nilpotent if it has a central series of finite length. Equivalently,
the lower central series G0=G , Gi+1= [ Gi , G ] reaches the trivial subgroup after
finitely many steps[9]. In other words, iterated commutators eventually
exhaust G. Equivalently (for finite G), G is the direct product of its Sylow
Tilka Manjhi Univ.)
Unit 1: Groups – Homomorphisms, Actions, Series, and
Special Classes
Group Homomorphisms
A group homomorphism φ: G→H is a map between groups that preserves
the group operation. Equivalently, φ satisfies φ(e_G)=e_H and
φ(g₁g₂)=φ(g₁)φ(g₂) for all g₁,g₂∈G[1]. The kernel Ker(φ)={g∈G: φ(g)=e_H}
is a normal subgroup of G, and the image Im(φ)=φ(G) is a subgroup of H.
Key results include the First Isomorphism Theorem: G/Ker(φ)≅Im(φ) (so φ
induces an isomorphism of quotient and image). For example, the
determinant map det: GL(n,ℝ)→ℝ^* is a homomorphism with kernel SL(n,ℝ)
(matrices of determinant 1). Proofs of the isomorphism theorem use the
universal property of quotients: the kernel being normal ensures well-
definedness of the quotient map[2].
Group Actions
A group action of G on a set X is a map G × X → X , denoted ( g , x ) ↦ g ⋅ x ,
satisfying e ⋅ x =x and g ⋅ ( h ⋅ x )=( g h ) ⋅ x. Equivalently, each g∈G gives a
permutation of X and the map g↦(x↦g·x) is a homomorphism G→Sym(X).
The orbit of x is Orb_G(x)={g·x: g∈G} and the stabilizer is
Stab_G(x)={g∈G: g·x=x}. The Orbit–Stabilizer Theorem states that for finite
G,
$$|G| = |{\rm Orb}_G(x)|\,|{\rm Stab}_G(x)|,$$
so the orbit size divides |G|[3]. For example, a group acting on itself by
conjugation (g·x=gxg^{-1}) has orbits the conjugacy classes, and
Stab(x)=C_G(x), the centralizer. Group actions link to counting arguments
(e.g. counting colorings or symmetries) via Burnside’s Lemma.
Sylow Theorems
Let |G|=p^e m with p∤m. A Sylow p-subgroup is a subgroup of order p^e.
The three Sylow Theorems assert:
- Sylow I: G has at least one Sylow p-subgroup.
- Sylow II: All Sylow p-subgroups are conjugate in G.
- Sylow III: The number n p of Sylow p-subgroups satisfies n p ≡1(mod p) and
divides m.
, For instance, a group of order 21 (3·7) must have a unique Sylow-7 subgroup
(so it is normal) and either 1 or 3 Sylow-3 subgroups (each of order 3)[4]. A
typical proof (via group actions on left cosets) shows existence by counting
fixed points, and uniqueness up to conjugacy by action on the set of Sylow
subgroups. These theorems constrain possible group structures: e.g. if
n_p=1 then the Sylow p-subgroup is normal.
Normal and Subnormal Series; Composition Series
A normal series of G is a chain of subgroups
{1 }=N 0 ⊲ N 1 ⊲ ⋯ ⊲ N k =G ,
where each N i ⊲N i+1. A subnormal series only requires each N i normal in G (or
each normal in the next). A series is refinement of another if the chain is
strictly longer. A composition series is a subnormal series with each factor
N i+ 1 /N i simple (having no nontrivial normal subgroups)[5]. By a standard
argument using Zorn’s Lemma (or simply induction on |G|), every finite
group has a composition series.
Jordan–Hölder Theorem: Any two composition series of a finite group G
have isomorphic factor groups up to order. In particular, the list of simple
factors (Jordan–Hölder factors) is uniquely determined (up to permutation
and isomorphism)[6]. For example, S3 has a composition series 1 ⊲C 2 ⊲S 3 with
factors C 2 and C 3. A different series is not possible (since one of 2 or 3 must
appear). The JH theorem is proved by refining and comparing series using
Schreier’s theorem.
Solvable and Nilpotent Groups
A group G is solvable if it has a subnormal series whose factor groups are all
abelian[7]. Equivalently, its derived series G(0 )=G , G(1)= [ G , G ] , G(n +1)= [ G (n) , G(n )]
eventually reaches the trivial subgroup[7]. For finite G, this is also equivalent
to having a composition series whose factors are cyclic of prime order[8].
Examples include all abelian and all p-groups. Notably, S3 (of order 6) is
solvable, while A5 (of order 60) is not, since A5 has nonabelian composition
factors. The commutator subgroup [G,G] (generated by all commutators
−1 −1
x y x y ) measures how nonabelian G is. G/[G,G] is the maximal abelian
quotient; iterating commutators gives the derived series mentioned above.
Solvability of G⇔G/[G,G] abelian, etc.
A group G is nilpotent if it has a central series of finite length. Equivalently,
the lower central series G0=G , Gi+1= [ Gi , G ] reaches the trivial subgroup after
finitely many steps[9]. In other words, iterated commutators eventually
exhaust G. Equivalently (for finite G), G is the direct product of its Sylow
