Math 361, Spring 2014, Assignment 8

Carefully define the following terms, then give one example and one non-example of each:

1. Meet (of two subgroups of a group $G$).
2. Join (of two subgroups).
3. Product (of two subgroups).
4. Subnormal series.
5. Factors (of a subnormal series).
6. Isomorphic (subnormal series).
7. Refinement (of a subnormal series).
8. Saturated (subnormal series).
9. Composition series.
10. Composition factors.

Carefully state the following theorems (you need not prove them):

1. Lemma relating joins to products.
2. First isomorphism theorem.
3. Second isomorphism theorem.
4. Third isomorphism theorem.
5. Jordan-Hölder Theorem.

Solve the following problems:

1. Section 34, problems 3 and 5.
2. Section 35, problems 1, 3, and 7.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Meet (of two subgroups of a group $G$).

   Let $H,K$ be subgroups of $G$.

   The meet of $H$ and $K$, written $H\wedge K$, is the intersection $H\cap K$. This is again a subgroup of $G$, and $H\wedge K$ is the largest subgroup contained in both $H$ and $K$.

   Example:

   In $\mathbb{Z}_{12}$: $$\langle 4\rangle\wedge\langle 6\rangle = \langle 0\rangle$$ $$\langle 2\rangle\wedge\langle 3\rangle = \langle 6\rangle$$ $$\langle 3\rangle\wedge\langle 4\rangle = \langle 0\rangle$$

   Non-example:

   $$\langle 4\rangle\wedge\langle 6\rangle\neq\langle 2\rangle$$

2. Join (of two subgroups).

   The join of $H$ and $K$, written $H\vee K$, is the subgroup generated by $H\cup K$. Observation: $H\vee K$ is the smallest subgroup contained in both.

   Example:

   In $\mathbb{Z}_{12}$: $$\langle 4\rangle\vee\langle 6\rangle = \langle 2\rangle$$ $$\langle 2\rangle\vee\langle 3\rangle = \langle 1\rangle$$

   Non-example:

   $$\langle 4\rangle\vee\langle 6\rangle\neq\langle 0\rangle$$

3. Product (of two subgroups).

   The product of $H$ and $K$, written $HK$, is given by: $$HK=\lbrace hk | h\in H, k\in K\rbrace$$.

   Note: If either H or K is normal, then the product is a subgroup and $HK=H\vee K$.

4. Subnormal series.

   Let $G$ be a group. A subnormal series for $G$ is a chain of normal subgroups, denoted: $$H_1\unlhd H_2\unlhd H_3\unlhd\cdots\unlhd H_n$$.

5. Factors (of a subnormal series).

   The quotient groups $H_i / H_{i-1}$ are called the factors of the series.

6. Isomorphic (subnormal series).

   Two subnormal series: $$H_0\unlhd\cdots\unlhd H_n$$ $$K_0\unlhd\cdots\unlhd K_m$$ are isomorphic if they have isomorphic factors.

   Example:

   In $\mathbb{Z}_{12}$: $$\langle 0\rangle\subseteq\langle 4\rangle\subseteq\langle 2\rangle\subseteq\mathbb{Z}_{12}$$ $$\langle 0\rangle\subseteq\langle 6\rangle\subseteq\langle 3\rangle\subseteq\mathbb{Z}_{12}$$ are isomorphic.

7. Refinement (of a subnormal series).

   A refinement of a subnormal series is any other subnormal series obtained by inserting new subgroups into the chain.

   Example

   In $\mathbb{Z}_{12}$: $$\langle 0\rangle\subseteq\langle 4\rangle\subseteq\langle 2\rangle\subseteq\mathbb{Z}_{12}$$ is a refinement of $$\langle 0\rangle\subseteq\langle 2\rangle\subseteq\mathbb{Z}_{12}$$

8. Saturated (subnormal series).

   A subnormal series is saturated if it has no refinements.

   Example

   In $\mathbb{Z}_{12}$: $$\langle 0\rangle\subseteq\langle 4\rangle\subseteq\langle 2\rangle\subseteq\mathbb{Z}_{12}$$

   Non-example

   In $\mathbb{Z}_{12}$: $$\langle 0\rangle\subseteq\langle 2\rangle\subseteq\mathbb{Z}_{12}$$

9. Composition series.

   A composition series for $G$ is a saturated subnormal series.

   Example

   In $\mathbb{Z}_{12}$: $$\langle 0\rangle\subseteq\langle 6\rangle\subseteq\langle 2\rangle\subseteq\mathbb{Z}_{12}$$

   Non-example

   In $\mathbb{Z}_{12}$: $$\langle 0\rangle\subseteq\langle 3\rangle\subseteq\mathbb{Z}_{12}$$

10. Composition factors.

    The composition factors of $G$ are the factors of any composition series.

    Example:

    The composition factors of $\mathbb{Z}_{12}$ are $\lbrace\mathbb{Z}_2,\mathbb{Z}_2,\mathbb{Z}_3\rbrace$.

    Non-example:

    The composition factors of $\mathbb{Z}_{12}$ are not$\lbrace\mathbb{Z}_2,\mathbb{Z}_3\rbrace$, you need to show the repetition of the $\mathbb{Z}_2$ factor.

Theorems:

1. Lemma relating joins to products.

   If either $H$ or $K$ is normal, then $HK$ is a subgroup and $HK= H\vee K$

2. First isomorphism theorem.

   aka: The fundamental theorem of homomorphisms

   Suppose $\phi :G\rightarrow G'$ is a homomorphism. Then there exists a unique monomorphism $\phi^* :G/ker\phi \rightarrow G'$. In particular $im\phi\simeq G/ker\phi$.

3. Second isomorphism theorem.

   Suppose $H$ and $K$ are subgroups of $G$ and $K$ is normal. Then: $$\frac{HK}{K}\simeq\frac{H}{H\wedge K}$$

4. Third isomorphism theorem.

   Suppose $K\subseteq H\subseteq G$ and both $H$ and $K$ are normal in $G$. Then, $$\frac{G/K}{H/K}\simeq\frac{G}{H}$$

5. Jordan-Hölder Theorem.

   Any two composition series for $G$ are isomorphic.