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Algebraic Sturcture 101

Group

Group

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

    1. Identity: exists eG such as ea=ae=a
    2. Inverse: for each aG, exists a bG such as ab=ba=e
    3. Associative: for all a,b,cG, 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 abb1=cbb1 a(bb1)=c(bb1) ae=ce a=c

    1. 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.

    2. Shoe and socks property: (ab)1=b1a1

Subgroup

  1. Subgroup: If G is a group and HG 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 an, n>1

  3. give an example of a group that is not cyclic: D4

  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 gn=e

  6. One-step subgroup test: Let G be a group and H a nonempty subset of G, ab1H when a,bH, then H is a subgroup of G

  7. Two-step subgroup test:

    1. abH
    2. a1H

Cyclic Group

  1. Center: Z(G)=aG|ag=ga$ for all $gG

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

    1. Since eg=ge=g for gG,

    2. Let a1,a2Z(G), since a1a2g=a1ga2=g(a1a2), a1a2G

    3. Let aZ(G), ag=ga a1ag=a1ga g=a1ga ga1=a1g Thus, a1Z(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)=gG|ag=ga

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

    1. Since ea=ae, eC(a)

    2. Let g1,g2C(a), since g1g2a=g1ag2=ag1g2, g1g2C(a)

    3. Let gC(a), and g1 be the inverse of g in G, ag=ga g1agg1=g1gag1 g1a=ag1

  5. Criterion for ai=aj: If a has infinite order, ai=aj if only if i=j. If a has finite order, then a=e,,an1

  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 ak=ad and |ak|=nd

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

    • Every subgroup of cyclic group is cyclic
    • If |a|=n, the order of any subgroup of a is a divisor of n
    • For each positive divisor k of n, the group a has exactly one subgroup of order k, an/k

Permutation Group

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

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

  3. Symmetric group: Sn= permutations of 1,2,,n, |Sn|=n!

  4. Disjoint cycles commute

  5. multiply the following two permutations in cycle notation

  6. Write the permutation as a product of transpositions: (12)(23)(n1n)=(12)(13)(1n)

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

  8. Always even or always odd theorem for permutations: If αSn, and α=t1tk=s1sl, then k and l are both even or odd

  9. If An is the subset of even permutations, then |An|=|Bn|=n!2, |An| is a subgroup

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