Pure Mathematics E1: Cosets & Normal Subgroups

52 flashcards · Mathematics · by lakesare

What does frac(x) stand for?
The fractional part of a real number.
What's the formula for frac(x)?
frac(x) = x − ⌊x⌋
Visually imagine frac(x).
Either 0, or distance from x to the greatest integer down.
What does +1 mean?
The binary operation +1 is defined on the interval [0, 1)
What group properties is it easy to check from the Cayley table? Cayley table for the (G, ×20) group.
G1 Closure G2 Associativity G3 Identity G4 Inverses
Name 2 standard groups under +n and ×n.
(Zn, +n) (Un, ×n)
What's the relationship between Un and Zn*.
Un == Zn* when n is prime.
How would you prove that a group table is like a sudoku? A Cayley table of a group is called a group table.
How can a permutation be even or odd? Permutation is a function from S to S.
It's about the number of transpositions that the permutation can be represented in.
Name 2 steps to determine the parity of the permutation just by looking at it?
Determine the parity of each cycle (n – 1) Add these parities (as if these are integers)
Name 2 famous permutation groups.
Sn - symmetric group of degree n Sn = { all permutations of { 1... n } }An - alternating group of degree n An = { all even permutations of { 1... n } }
Plane figure VS solid figure? A figure in R2 is any subset of R2, such as a triangle, a square, a rectangle or a line. Similarly, a figure in R3 is any subset of R3, such as a tetrahedron or a cuboid.
Plane figure - 2d Solid figure - 3d
What does "the line 14" mean?
The line through the vertices at locations 1 and 4.
Name 2 classic groups that matrices form.
(Mm,n, +) (invertible Mn,n, ×) The second of these groups is called the general linear group of degree n (over the real numbers), and is denoted by GL(n).
Name 2 things that GL stands for.
OpenGL - Open Graphics Library OpenGL Shading Language (GLSL) is the principal shading language for OpenGL.GL(n) - General Linear group of degree n GL(n) = the set of all invertible n × n matrices with real entries forms a group under matrix multiplication.
Visually imagine g ◦ f. When both permutations are expressed in a two-line form.
What does GL(n) vs SL(n) stand for?
GL - General Linear group of degree n SL - Special Linear group of degree n
Visually imagine GL(n) (General Linear group of degree n). GL - General Linear group of degree nSL - Special Linear group of degree n
All invertible matrices.
Visually imagine SL(n) (Special Linear group of degree n). GL - General Linear group of degree nSL - Special Linear group of degree n
All matrices with determinant = 1.
Name 4 subgroups of the General Linear group GL(n).
Special Linear group SL(n) = { all det(A) = 1 matrices } ◣ = { all invertible lower triangular matrices } ◥ = { all invertible upper triangular matrices } ⟍ = { all invertible diagonal matrices } The group GL(2) has very many more subgroups than the four above.
Visually imagine element x: with finite order with infinite order x belongs to group (G, ◦)
x has finite order (4) x has infinite order
How does the Lagrange's theorem relate to the order of the particular element? Joseph-Louis Lagrange (1736 - 1813)
Order of x always divides the order of G.
What's common between: composition of even/odd permutations composition of direct/indirect symmetries
Both behave like the addition of integers (odd permutations, indirect symmetries == odd integers)
Name 3 steps to check whether φ you found is an isomorphism from G to H.
Determine φ(x) Determine φ(y) Check that φ(x ◦ y) = φ(x) ∗ x(y)
[Isomorphism classes] What does this table show?
All isomorphism classes per each group order. This table shows all isomorphism classes for groups of orders 1 to 8. For each isomorphism class, the table gives one or two standard groups in the class.There is a row for each different isomorphism class, so the table shows that for each of the orders 1, 2, 3, 5 and 7 there is just one isomorphism class, whereas for each of the orders 4 and 6 there are two isomorphism classes, and for order 8 there are five isomorphism classes.
[Isomorphism classes] What group does Cn denote?
Cn = any cyclic group of order n
[Isomorphism classes] What group does ℤn denote?
ℤn = (ℤn, +n)
[Isomorphism classes] What group does Un denote?
Un = (Un, ×n)
[Isomorphism classes] What group does V denote?
V = S(▭, ◦) = Klein four-group
[Isomorphism classes] What's notable about prime-order groups?
ALL prime-order groups are isomorphic to each other (and isomorphic to (ℤp, +p)).
Write these in concise multiplicative notation. (G, ◦) - _______________ a ◦ b - _______________
(G, ◦) - G a ◦ b - ab
[Cosets] How can we rewrite gH?
g ◦ H
[Cosets] How would you denote in this situation: left coset right coset
left coset - gH right coset - Hg
[Cosets] Expand aH.
aH = a ◦ H = a ◦ { e, r } = { a ◦ e, a ◦ r }
[Cosets] What's a good analogy about obtaining a coset of a subgroup? A left coset of a subgroup H in a group G
We shift the subgroup.
[Cosets] What can we say about all left cosets of a subgroup? Groups partitioned into left cosets of a subgroup:(a) S(△) with subgroup { e, s }(b) Z*7 with subgroup { 1, 2, 4 }
All left cosets of a subgroup form a partition of a group.
[Cosets] What's the overall method of proving that left cosets partition the set G. Groups partitioned into left cosets of a subgroup:(a) S(△) with subgroup { e, s }(b) Z*7 with subgroup { 1, 2, 4 }
Define relation ~ on G: x ~ y if y ∈ yH Show ~ is an RST relation RST relations partition their domain - then so does ~! The overall method of the proof is that we define a particular relation on the set G, prove that it is an equivalence relation, and prove that its equivalence classes are the left cosets of H in G. It then follows from Theorem A16 that these left cosets form a partition of G.
[Cosets] How could the partitions of S(□) group into left cosets right cosets looks?
The partitions of S(□) into (a) left and (b) right cosets of { e, r }
[Cosets] When are the left and right cosets of the group the same?
In many cases, but e.g. when G is abelian.
[Cosets] Given all left cosets of H, how would you go about finding all right cosets of H? The partitions of S(□) into (a) left and (b) right cosets of { e, r }
Take –1 of each el.
[Index of a subgroup] What's the index of the subgroup H in a group G? The partitions of S(□) into (a) left and (b) right cosets of { e, r }
Number of distinct left cosets (or, equivalently, the number of distinct right cosets).
[Index of a subgroup] How would you calculate the index of the subgroup H in a group G? The partitions of S(□) into (a) left and (b) right cosets of { e, r }
|G|/|H|
[Normal subgroup] What's a normal subgroup?
A subgroup is normal when { left cosets } = { right cosets }. You saw that the partition of a group into left cosets of a particular subgroup may be different from its partition into right cosets of the same subgroup. You saw that if the group is abelian then the two partitions are the same. There are also some non-abelian groups and subgroups for which the two partitions are the same.
What does "H ◁ G" denote in some texts?
H is a normal subgroup of G.
What 2 normal subgroups does every group have?
trivial subgroup { e } whole group G Because "normal" subgroup means a "set-wise abelian" subgroup. Coincidentally these are the same as the "subgroups that a group always has"!
[Index of a subgroup] Visually imagine a subgroup of index 2.
[Normal subgroup] How would you show that "if index is 2 ➝ this is a normal subgroup"?
Because the first coset is H itself.
[Normal subgroup] How would you show An is always a normal subgroup of Sn?
Sn has the order n! An has the order n!/2
[Normal subgroup] How do we know that all direct symmetries of S(tet) is a normal subgroup?
[Normal subgroup] What does this express in notation? (∀g)(g ∈ G ➝ gH = Hg)
That H is a normal subgroup.
[Normal subgroup] When can we say just "cosets" instead of "left cosets" and "right cosets"?
We say just "cosets" for normal subgroups. Remember that for normal subgroups we refer simply to cosets rather than left cosets or right cosets, because for normal subgroups left cosets and right cosets are the same.
Degree of the symmetric group VS index of the subgroup?
degree - number of elements that each f maps index