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Finite Abelian Groups and Solvable Groups

Finitely generated groups

See this video and these notes.

Definition: Let $G$ be a group and let $T={g_i: i\in I}$ be a (not necessarily finite) collection of elements of $G$. The subgroup of $G$ generated by $T$ is the smallest subgroup of $G$ that contains $T$. We also say that $G$ is generated by $T$.

Examples:

  • Let $G=S_{n}$ and let $T$ be the set of transpositions. Then the subgroup generated by $T$ is all of $G$.

  • Let $G$ be any group and let $T={g}$ for some element $g\in G$. Then the subgroup of $G$ generated by $T$ is the cyclic subgroup $\langle g\rangle$.

  • Let $G$ be the dihedral group with $2n$ elements and let $T={R,S}$ where $S$ is a reflection and $R$ is a generator of the rotation subgroup. Then $T$ generates $G$.

Definition: A group $G$ is said to be finitely generated if there exists a finite set $T$ that generates $G$.

Examples:

  • Any finite group is finitely generated because it is generated by the finite set consisting of all its elements.

  • The integers are finitely generated because they are generated by ${1}$.

  • The group of euclidean motions preserving a lattice (a “wallpaper group”) is finitely generated because it is generated by two fundamental translations together with finitely many rotations/reflections.

  • The rational numbers with addition are not finitely generated. (See Example 13.2).

The Fundamental Theorem of Finite Abelian Groups

See this video and these notes.

Theorem: Let $G$ be an abelian group with $n$ elements. Then $G$ is isomorphic to a product \(G = \mathbb{Z}_{p_{1}^{n_{1}}}\times\mathbb{Z}_{p_{2}^{n_{2}}}\times \cdots\times \mathbb{Z}_{p_{k}^{n_{k}}}\) where the $p_{i}$ are (not necessarily distinct) prime numbers. Note that in this case \(n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}.\)

The proof has four main steps.

See this video and these notes

  1. If $G$ is a finite abelian group of order $n$, and $p$ is a prime that divides $n$, then $G$ contains an element of order $p$.

  2. Let $G$ be a finite group. If every element of $G$ has order $p^{k}$ for some $k$, where $p$ is a prime, then the order of $G$ is a power of $p$. Conversely, if every element of $G$ is a power of $p$ for some prime, then the order of $G$ is a power of that prime. Such a group is called a finite abelian $p$-group.

See this video and these notes

  1. If $G$ is a finite abelian $p$-group, and $g\in G$ has maximal order among all orders of elements of $G$, then $G$ is isomorphic to $\langle g\rangle \times H$ for some subgroup \(H\) of $G$.

See this video and these notes.

  1. If $G$ is a finite abelian group of order $n=ab$ where $\mathrm{gcd}(a,b)=1$ then $G$ is the internal directo product of its subgroups $G_{a}$ and $G_{b}$ where $G_{a}$ is the subgroup of all elements of order dividing $a$ and $G_{b}$ is the subgroup of all elements of order dividing $b$.

To put these parts together:

  • use (3) to split $G$ into a product of subgroups $G_{i}$ where every element of $G_{i}$ has order a power of $p_{i}$ for distinct primes $p_{i}$.

  • by (1), each factor $G_{i}$ is a finite abelian $p_{i}$-group, so has order a power of $p_{i}$.

  • by (2), split each factor into a product of cyclic groups, each with order a power of $p_{i}$.

  • any cyclic group with $p_{i}^s$ elements is isomorphic to \(\mathbb{Z}_{p_{i}^{s}}\).

Examples:

  • Suppose that $G$ is abelian of order $12$. Then $G$ is isomorphic to one of the following groups:

    • \(\mathbb{Z}_{12}\).
    • \(\mathbb{Z}_{3}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}=\mathbb{Z}_{2}\times\mathbb{Z}_{6}\).
  • More generally if $G$ is an abelian group of order $n$, then the possible isomorphism classes ocorrespond to sequences of integers \(d_1|d_2|\cdots | d_k\) where $d_1 d_2 \cdots d_k=n$.

The Fundamental Theorem of Finitely Generated Abelian Groups

Theorem: Let $G$ be a finitely generated abelian group. Then $G$ is isomorphic to a direct product \(G=\mathbb{Z}^{k}\times G_{\mathrm{tor}}\) where $G_{\mathrm{tor}}$ is a finite abelian group consisting of all elements of finite order in $G$ (and hence is a product of abelian cyclic $p$-groups as above.)

Solvable groups

See this video and these notes.

Definition: A group is called solvable if there is a sequence of subgroups \(G\supset H_{1}\supset H_{2}\supset\cdots\supset\{0\}\) where each subgroup $H_{i+1}$ is normal in $H_{i}$ and if the quotients $H_{i}/H_{i+1}$ are abelian.

Examples

  • Abelian groups are solvable

  • $S_3$ and $S_4$ are solvable.

  • The dihedral groups are solvable.

  • $S_{5}$ is not solvable.