Review Sheet Math 581

[n] means the topic is found on acetate n.

Definitions

Group [2]

Identity element

Inverse element [2]

Abelian group [20]

nonAbelian group [20]

Examples:

Groups of permutations [3]

Zn under + [25]

Cyclic groups [29 to 33.2]

Groups of small order, like 1, 2, 3, 4, 5 [39, 40, 47, 48, 49]

Cosets, left and right [41]

Examples [41, 42, 43]

Subgroup (g) generated by an element g. [29]

Cyclic group [32]

Order of a group [38]

Order of a group element [38]

Index (G: H) of a subgroup H in a group G. [46.1]

Function

Well-defined

Onto

One-to-one

Permutation of a set [3]

n-cycle [8]

transposition (2-cycle) [7]

even permutation [8.9]

odd permutation [8.9]

Isomorphism of groups [24]

Subgroup [27.0, 27.1]

Homomorphism of groups [57]

Examples [58 to 60.2]

Kernel of a homomorphism [61.3]

Normal subgroup [66]

Quotient group G/H of a group G over a normal subgroup H. [67 to 70]

Symmetry of a geometric figure [75]

Examples [76, 78]

Theorems--with proofs

The set of all permutations of a set S with multiplication equal to composition of functions is a group. [4.0 to 4.5]

Every permutation is a product of transpositions [7 to 8.0]

The identity element in a group is unique [20]

The inverse of a given group element is unique [21]

(a')' = a [21]

(ab)' = b'a' [22]

If ab = ac, then b = c [22]

If ba = ca, then b = c [22]

If a is a fixed element of a group G, then the function F which maps g in G to ga, is a permutation of G. [ 23]

If H is a subgroup of G, then the identity element of g is in H. [27.1]

If H is a nonempty subset of a group G with the property that for every h, k in H, hk' is in H, then H is a subgroup of G. [28]

The set of all powers of a fixed element g is a subgroup. [29]

A cyclic group is Abelian [30]

If gk = gm for different exponents k and m, then (g) is a finite cyclic group and the number of elements in (g) is equal to the least positive integer n for which gn = e. [31]

If H is a subgroup of G, then

If a is in H, then aH = Ha = H;

If a and b are in G, then aH and bH are either disjoint or equal (as subsets of G) (Same for Ha and Hb.)

All cosets, left and right have the same number of elements. [44 to 46.0]

LaGrange's Theorem [46.1]

If F is a homomorphism from G into C, then

F maps the identity of G to the identity of C [61.0],

F(a') = F(a)' for every a in G [61.0]

The kernel of a homomorphism is a normal subgroup. [61.4]

A homomorphism is one-to-one if and only if its kernel is {e}. [65]

If H is a normal subgroup of a group G, the mapping F from G onto G/H given by F(a) = aH is a homomorphism with kernel H. [71.0, 71.1]

Theorems--no proofs

Every permutation of a finite set can be written as a product of disjoint cycles [5.5]

Every permutation is either even or odd, but not both [8.9 to 14]

Every cyclic group is either infinite and isomorphic to Z, or is finite and, if it has n elements, is isomorphic to Zn. [30 to 33.0]

Every subgroup of a cyclic group is cyclic [34 to 35]

Know what are all the subgroups of a cyclic group [36 to 37.1]

Every group of prime order p is isomorphic to Zp, + [49]

Every group of order 4 isomorphic to either Z4 or Z2 X Z2. [47, 48]

Every group of order 6 isomorphic to either Z6 or S3. [50 to 56.3]

If H is a subgroup of G, then aH = bH if and only if a'b is in H. [56.4]

If F is a homomorphism from G into C, then

F (ak) = F(a)k for every a in G and every integer k [61.0];

if H is a subgroup of G, then F(H) is a subgroup of C. [61.1]

If H is a subgroup of G and g is a fixed element of G, then g'Hg is a subgroup of G and is isomorphic to H. [61.2]

A normal subgroup is the kernel of some homomorphism. [66]

A symmetry of a polygon or polyhedron is completely determined by how it permutes the vertices. [77]

The set of symmetries of a geometric figure is a group with multiplication equal to composition of functions. [79, 80]

Calculations

Write any permutation as a product of disjoint cycles [5.5]

Calculate products using cycle notation [6]

Write any permutation as a product of transpositions [7 to 8.0]

Determine whether a permutation is even or odd [8.9]

Determine whether or not a given configuration of the fifteen puzzle can be reached from the initial configuration. [16 to 19]

Notation

e for identity element [2]

a' or a-1 for the inverse of a [2]

composition of functions, read left to right [3]

aF for F(a) [4.1, 4.2]

Two-row notation for permutations [5]

Cycle notation for permutations [5.0]

Notation for isomorphism [25]

(g) for the cyclic subgroup generated by g. [29]

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