Algebraic Sturcture 101

Group

Group

1. Group: A group is a non-empty set $G$ with binary operation satisfying:

   1. Identity: exists $e \in G$ such as $ea = ae = a$
   2. Inverse: for each $a \in G$, exists a $b \in G$ such as $ab = ba = e$
   3. Associative: for all $a, b, c \in G$, we have $a(bc) = (ab)c$

2. Properties:

   1. Uniqueness of the identity: There is only one identity element
   2. Cancellation in a group: If $ab=ac$ or $ba=ca$, $b=c$

   $$ ab = cb \ abb^{-1} = cbb^{-1} \ a(bb^{-1}) = c(bb^{-1}) \ ae = ce \ a = c $$

   3. Uniqueness of inverse in a group:

      Suppose $b, c$ is the inverse of $a$ in the group, since $ab = e, ac = e$, then $ab = ac$. By cancellation, $b = c$.

   4. Shoe and socks property: $(ab)^{-1} = b^{-1}a^{-1}$

Subgroup

1. Subgroup: If $G$ is a group and $H \subseteq G$ is a subset of $G$, then $H$ is a subgroup of a group G if it is closed under the same binary operation as $G$

2. Cyclic subgroup: of G generated by a: Every element in $G$ is equal to $a^n$, $n \gt 1$

3. give an example of a group that is not cyclic: $D_4$

4. Order of a Group: $|G|$, The number of elements of a group

5. Order of an Element: $|g|$, the smallest positive integer $n$ such that $g^n = e$

6. One-step subgroup test: Let $G$ be a group and $H$ a nonempty subset of $G$, $ab^{-1} \in H$ when $a, b \in H$, then $H$ is a subgroup of $G$

7. Two-step subgroup test:

   1. $ab \in H$
   2. $a^{-1} \in H$

Cyclic Group

1. Center: $Z(G) = {a \in G | ag = ga$ \text{ for all } $g\in G}$

2. Show that Z(G) is a subgroup of G

   1. Since $eg = ge = g$ for $g \in G$,

   2. Let $a_1, a_2 \in Z(G)$, since $a_1a_2 g = a_1ga_2 = g(a_1a_2)$, $a_1a_2 \in G$

   3. Let $a \in Z(G)$, $$ ag = ga \ a^{-1}ag = a^{-1}ga \ g = a^{-1}ga \ ga^{-1} = a^{-1}g $$ Thus, $a^{-1} \in Z(G)$

3. Centralizer: For $a$ in $G$, the centralizer of a in $G$, $C(a)$ is the set of all elements in $G$ that commute with $a$, $C(a) = {g \in G | ag = ga}$

4. Show that C(a) is a subgroup of G

   1. Since $ea = ae$, $e \in C(a)$

   2. Let $g_1, g_2 \in C(a)$, since $g_1g_2a = g_1ag_2 = ag_1g_2$, $g_1g_2 \in C(a)$

   3. Let $g \in C(a)$, and $g^{-1}$ be the inverse of $g$ in $G$, $$ ag = ga \ g^{-1}agg^{-1} = g^{-1}gag^{-1} \ g^{-1}a = ag^{-1} $$

5. Criterion for $a^i = a^j$: If $a$ has infinite order, $a^i = a^j$ if only if $i=j$. If $a$ has finite order, then $\langle a \rangle = {e, …, a^{n-1}}$

6. Property of gcd: If $d = gcd(a, b)$, there is integers $t$ and $s$ such as $d = ta+sb$

7. Let $G$ be a finite cyclic group of order $n$, if $d = gcd(k, n)$, then $\langle a^k \rangle = \langle a^d \rangle$ and $|\langle a^k \rangle | = \frac{n}{d}$

8. The fundamental theorem of cyclic groups (Theorem 4.3):

   • Every subgroup of cyclic group is cyclic
   • If $|\langle a \rangle| = n$, the order of any subgroup of $\langle a \rangle$ is a divisor of $n$
   • For each positive divisor $k$ of $n$, the group $\langle a \rangle$ has exactly one subgroup of order $k$, $\langle a^{n/k} \rangle$

Permutation Group

9. Permutation: a function from $A$ to $A$ that is one-to-one and onto

10. Permutation group: a set of permutations of $A$ that forms a group under function composition

11. Symmetric group: $S_n = $ permutations of ${1, 2, …, n}$, $|S_n| = n!$

12. Disjoint cycles commute

13. multiply the following two permutations in cycle notation

14. Write the permutation as a product of transpositions: $(12)(23)…(n-1n) = (12)(13)…(1n)$

15. Order of a permutation: The order of a permutation of a finite set is the least common multiple of the lengths of the cycles

16. Always even or always odd theorem for permutations: If $\alpha \in S_n$, and $\alpha = t_1…t_k = s_1…s_l$, then $k$ and $l$ are both even or odd

17. If $A_n$ is the subset of even permutations, then $|A_n| = |B_n| = \frac{n!}{2}$, $|A_n|$ is a subgroup