Chapter 9: Isomorphisms

Definitions and Examples

  • Isomorphic (同构)

    Two groups $(G, \cdot)$ and $(H, \circ)$ are isomorphic if there exists a one-to-one and onto map $\phi: G\to H$ such that the group operation is preserved; that is,
    $$\phi(a \cdot b) = \phi(a) \circ \phi(b)$$
    for all $a$ and $b$ in $G$. If $G$ is isomorphic to $H$, we write $G\cong H$.

  • Isomorphism

    The map $\phi$ is called an isomorphism.

  • Theorem 9.6

    Let $\phi: G\to H$ be an isomorphism of two groups. Then the following statements are true.

    1. $\phi^{-1}: H\to G$ is an isomorphism
    2. $|G|=|H|$
    3. If $G$ is abelian, then $H$ is abelian
    4. If $G$ is cyclic, then $H$ is cyclic
    5. If $G$ is a subgroup of order $n$, then $H$ is a subgroup of order $n$

  • Theorem 9.7

    All cyclic groups of infinite order are isomorphic to $\mathbb{Z}$

  • Theorem 9.8

    If $G$ is a cyclic group of order $n$, then G is isomorphic to $\mathbb{Z}_n$.

  • Corollary 9.9

    If $G$ is a group of order $p$, where $p$ is a prime number, then $G$ is isomorphic to $\mathbb{Z}_p$

  • Theorem 9.10

    The isomorphism of groups determines an equivalence relation on the class of all groups.
    Classifying all groups up to isomorphism; two groups are the same if they are isomorphic.

Cayley’s Theorem

  • Theorem 9.12 Cayley

    Every group is isomorphic to a group of permutations.

  • Left Regular Representation of $G$.

    The isomorphism $g\to \lambda_g$
    where $g\in G$, $\lambda_g(a)=ga$, $\bar{G}=\{\lambda_g:g\in G\}$ is the permutation group that is isomorphic with $G$

Direct Products

External Direct Products

  • Proposition 9.13

    Let $G$ and $H$ be groups. The set $G\times H$ is a group under the operation $(g_1, h_1)(g_2, h_2) = (g_1g_2, h_1h_2)$ where $g1, g2 \in G$ and $h_1, h_2 \in H$.

  • External Direct Product of $G$ and $H$ (外直积)

    The group $G\times H$

  • Theorem 9.17

    Let $(g, h) \in G\times H$. If $g$ and $h$ have finite orders $r$ and $s$ respectively, then the order of $(g, h)$ in $G\times H$ is the least common multiple of $r$ and $s$.

  • Corollary 9.18

    Let $(g_1, . . . , g_n) \in \prod G_i$. If $g_i$ has finite order $r_i$ in $G_i$, then the order of $G_i$ is the least common multiple of $r_1,\cdots, r_n$.

  • Theorem 9.21

    The group $\mathbb{Z}_m\times \mathbb{Z}n$ is isomorphic to $\mathbb{Z}{mn}$ if and only if $gcd(m,n) = 1$.

  • Corollary 9.22

    Let $n_1, \cdots,n_k$ be positive integers. Then
    $$\prod_{i=1}^k\mathbb{Z}_{n_i}\cong\mathbb{Z}_{n_1\cdots n_k}$$
    if and only if $gcd(n_i,n_j)=1$ for $i\ne j$

  • Corollary 9.23

    If
    $$m=p_1^{e_1}\cdots p_k^{e_k}$$
    where the $p_i$s are distinct primes, then
    $$\mathbb{Z}m\cong\mathbb{Z}{p_1^{e_1}}\times\cdots\times\mathbb{Z}_{p_k^{e_k}}$$

Internal Direct Products

  • Internal Direct Product (内直积)

    Let $G$ be a group with subgroups $H$ and $K$ satisfying the following conditions.

    1. $G=HK=\{hk:h\in H,k\in K\}$
    2. $H\cap K={e}$
    3. $hk=kh$ for all $k\in K$ and $h\in H$

    Then G is the internal direct product of H and K.

  • Theorem 9.27

    Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H\times K$.

  • Theorem 9.29

    Let $G$ be the internal direct product of subgroups $H_i$, where $i = 1, 2, \cdots, n$. Then $G$ is isomorphic to $\prod_iH_i$

Chapter 10: Normal Subgroups and Factor Groups

Factor Groups and Normal Subgroups

Normal Subgroups

  • Normal Subgroup of a group $G$ (正规子群)

    A subgroup $H$ of a group $G$ is normal in $G$ if $gH = Hg$ for all $g \in G$.
    A normal subgroup of a group $G$ is one in which the right and left cosets are precisely the same.

  • Theorem 10.3

    Let $G$ be a group and $N$ be a subgroup of $G$. Then the following statements are equivalent.

    1. The subgroup $N$ is normal in $G$
    2. For all $g\in G$, $gNg^{-1}\subset N$
    3. For all $g\in G$, $gNg^{-1}=N$

Factor Groups

  • Factor Group (因子群) or Quotient Group (商群) of $G$

    If $N$ is a normal subgroup of a group $G$, then the cosets of $N$ in $G$ form a group $G/N$ under the operation $(aN)(bN) = abN$.
    This group is called the factor or quotient group of $G$ and $N$.

  • Theorem 10.4

    Let $N$ be a normal subgroup of a group $G$. The cosets of $N$ in $G$ form a group $G/N$ of order $[G : N]$.

The Simplicity of the Alternating Group

  • Simple Groups (简单群)

    Groups with no nontrivial normal subgroups.

  • Lemma 10.8

    The alternating group $A_n$ is generated by 3-cycles for $n \geq 3$.

  • Lemma 10.9

    Let $N$ be a normal subgroup of $A_n$, where n ≥ 3. If $N$ contains a 3-cycle, then $N = A_n$

  • Lemma 10.10

    For $n \geq 5$, every nontrivial normal subgroup $N$ of $A_n$ contains a 3-cycle.

  • Theorem 10.11

    The alternating group, $A_n$, is simple for $n \geq 5$.

Chapter 11: Homomorphisms

Group Homomorphisms

  • Homomorphism (同态) between groups $(G, \cdot)$ and $(H, \circ)$

    A map $\phi: G\to H$ such that $\phi(g_1\cdot g_2)=\phi(g_1)\circ\phi(g_2)$ for $g_1, g_2 \in G$.

  • Homomorphic Image of $\phi$

    The range of $\phi$ in $H$.

  • Proposition 11.4

    Let $\phi: G_1\to G_2$ be a homomorphism of groups.

    1. If $e$ is the identity of $G_1$, then $\phi(e)$ is the identity of $G_2$;
    2. For any element $g \in G_1$, $\phi(g^{-1}) = [\phi(g)]^{−1}$;
    3. If $H_1$ is a subgroup of $G_1$, then $\phi(H_1)$ is a subgroup of $G_2$;
    4. If $H_2$ is a subgroup of $G_2$, then $\phi^{-1}(H_2) = \{g \in G_1 : \phi(g) \in H_2\}$ is a subgroup of $G_1$.
      Furthermore, if $H_2$ is normal in $G_2$, then $\phi^{-1}(H_2)$ is normal in $G_1$.

  • Theorem 11.5

    Let $\phi: G\to H$ be a group homomorphism. Then the kernel of $\phi$ is a normal subgroup of $G$.

The Isomorphism Theorems

  • The Natural or Canonical Homomorphism

    Let $H$ be a normal subgroup of $G$. $\phi:G\to G/H$, $\phi(g)=gH$
    is the natural or canonical homomorphism.

  • Theorem 11.10 First Isomorphism Theorem

    If $\psi : G \to H$ is a group homomorphism with $K = \textrm{ker} \psi$, then $K$ is normal in $G$. Let $\phi : G \to G/K$ be the canonical homomorphism. Then there exists a unique isomorphism $\eta : G/K \to \psi(G)$ such that $\psi = \eta\phi$.

  • Theorem 11.12 Second Isomorphism Theorem

    Let $H$ be a subgroup of a group $G$ (not necessarily normal in $G$) and $N$ a normal subgroup of $G$. Then $HN$ is a subgroup of $G$, $H \cap N$ is a normal subgroup of $H$, and
    $$H/H \cap N \cong HN/N$$

  • Theorem 11.13 Correspondence Theorem

    Let $N$ be a normal subgroup of a group $G$. Then $H \mapsto H/N$ is a one-to-one correspondence between the set of subgroups $H$ containing $N$ and the set of subgroups of $G/N$.
    Furthermore, the normal subgroups of $G$ containing $N$ correspond to normal subgroups of $G/N$.

  • Theorem 11.14 Third Isomorphism Theorem

    Let $G$ be a group and $N$ and $H$ be normal subgroups of $G$ with $N \subset H$. Then
    $$G/H\cong\frac{G/N}{H/N}$$