**22/5**Hall's theorem about subgroups of solvable groups.

**20/5**Fitting subgroups of a group: definition and basic properties. Relations between $F(G)$, $Z(G)$ and $\Phi(G)$. Fitting subgroup in solvable groups. Fitting series. Examples ($S_4$). Properties of Sylow subgroups and $G'$ when $F(G)$ is cyclic.

**13/5**Characterization of nilpotent groups as product of their Sylow subgroups. Minimal normal subgroups of nilpotent groups. Invariant series, solvable groups: definition and first properties.

**6/5**Nilpotent groups: ascending and descending central series. Nilpotence class: examples with $c = 0, 1, 2$. Growth of the normalizer of a subgroups of a nilpotent group. Maximal subgroups of nilpotent groups. Finite $p$-groups are nilpotent.

**4/5**Derived group: examples and basic properties. Central chains of subgroups.

**29/4**Free abelian groups: uinversal property of lifting a surjective group homomoprhism (projectivity of free groups). Injective and divisible groups.

**27/4**How describing the cyclic structure of a finitely generated abelian group once given a finite presentation. Smith formof a matrix over integers.

**24/4**Exercises.

**22/4**Abelian, finitely generated groups: descrption of their structure as sum of cyclic groups. Torsion subgroup. Examples.

**15/4**Subgroups of finitely generated groups. Frattini subgroup: properties and examples. Relations with the existence of maximal subgroups. Frattini argument.

**13/4**Classification of groups of order $12$. Sets of generators of groups. Finitely generated groups: definition and examples.

**27/3**On the normalizer of the intersection of two Sylow-subgroups. On the simplcity of groups of oder $p^nq$. Growth of normalizer of subgroups of finite $p$-groups. Simple groups of order $60$.

**25/3**Sylow groups of subgropus and of quotients of a given group $G$. The abelina case for Sylow theorems. Groups of order $pq$, $p^2q$. Simplicity in groups with subgroups of index $2,3,4$.

**18/3**Sylow theorems.

**16/3**$A_n$ is the only normal subgroups of $S_n$, for $n \neq 4$. Automorphisms of $S_n$ are all inner for $n \neq 6$. Subgroups of $S_n$ of index $n$.

**11/3**The quozient of a semidirect product. Simplicity of $A_n$ for $n \geq 5$.

**9/3**Construction of the semidirect product. Examples of groups of order $6$, dihedral groups $D_n$ and $D_{\infty}$. A criteria to establish when two semidirect products $H \rtimes_{\varphi} K$ and $H \rtimes_{\psi} K$ are isomorphic: examples. Classification of groups of order $pq$. Groups of order $8$.

**4/3**Jordan-Holder program for the classification of finite groups. Normal series, composition series and factors: examples. Direct product of groups. Semidirect product: definition.

**2/3**Normalizer and centralizer of a subgroup and their relation with the conjugacy relation among subgroups (particular case of a $n$-cycle in $S_n$). The biggest normal subgroup of a subgroup. Authomorphisms of a subgroup and their relations with the inner authomorphisms of a group $G$.

**27/2**On the cardinality of a finite group containing at least two non conjugate involutions. The example of $D_5$. Conjugation in $S_n$ ($\S$. 2.3 MachÃ¬): cyclic structure of permutations.

**22/2**Permutable subgroups. Characterization of groups with at most one subgroup of fixed order (cyclic groups). Homomorphism theorems. Conjugation among elements. Center of $G$ and centralizers of elements. Class equation and some applications: $Z(G) \neq \{1\}$, gorups of order $p^2$ are abelian, invetibility of Lagrange theorem in finite $p$-groups.

**20/2**Number of generators of a group of cardinality $n$ is $\leq log_2(n)$. Homomorphisms and authomorphisms of groups: some examples. Maximum number of non isomorphinc groups of cardinality $n$.

**18/2**Group: definition and basic properties. Subgroups: intersection and union, characterization. Cyclic groups and their subgroups. Symmetric group, dihedral groups. Lagrange theorem on the cardinality of subgroups of finite groups. Cosets of a subgroup.