MAT344H1 Lecture Notes - Lecture 2: Complement Graph, Mountaineering, If And Only If

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19 Sep 2018

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Suppose g = (v, e) is a graph, let f: v->v defines an isomorphism f: g-> g. then we call f an automorphism. Def: the degree of a vertex v in v written deg(v) or d(v) is the number of edges ending at v. Proposition: if f:g1->g2 is an isomorphism, then a and f(a) have the same degree, and g1 and g2 have the same number of degree n vertices for each n. Find a graph theoretic property of g, not shared by g2, ex. Number of degree n vertices, connectiveness, can delete 1 edge to disconnect, etc. Def: a subgraph g=(v,e) of h=(vh, eh) is an inclusion v in vh and e in eh s. t. If v1 and v2 in v are adjacent whenever v1 and v2 in vh are adjacent, call g an induced subgraph. If f: g1->g2 is an isomorphism, and h is the induced subgraph of g1 consisting of all vertices adjacent to v in.

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These questions are asking about subrgoups, groups, isormorphic.

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SAMPLE MIDTERM 1, MATH 120A

(1) 1. Write the denition of group.

2. Give an example of a group of order 2011.

3. Give an example of a non-abelian group with 12 elements.

(2) Let (G,*) and (H . ) be groups.

1. Dene what it means for G and H to be isomorphic.

2. Assume that G has exactly two solutions to the equation x^2 =e.

Prove, directly from the denition in (1), that if G and H areiso-

morphic, then H has exactly two solutions to the equation x^2 =e.

3. Prove or disprove: (R, +) is isomorphic (R+, . ). Here R+ ={x is subset of R : x>0}.

4. Prove or disprove: (R, +) is isomorphic (R*, . ). Here R* ={x is subset of R : x is not 0}

(3) Let G be the group Z_20, with addition modulo 20.

1. Draw the subgroup diagram of G, giving a generator for eachsubgroup

2. Find the subgroup H =< 16 > of G.

3. Find a subgroup K of U_20 isomorphic to H.

(4) Let (G, . ) be a group and let H be a subgroup of H. Fix anelement a of

G and let K = { aha^-1 : h is subset of H}

1. Prove that K is a subgroup of G.

2. Prove that the map psi : H -> K, h -> aha^-1 isan isomorphism of groups.

(5) Let (G, . ) be a group and let a be an element of G.

1. Dene the order of a.

2. Give an example of an element of order 4 in GL(2,R).

3. Find the order of 16 in Z_20.

1. Let V be an inner product space. Prove the Pythagoreantheorem: if v and w E V are orthogonal then ||v+w||^2 =||v||^2 + ||w||^2. (Suggestion: start with ||v+w||^2 = (v+w,v+w).)

2. Let V=P2, the polynomials of degree less than/equal to 2 withcoefficients in R, and let (.,.): VxV -> R be the map

(p,q) = p(0)q(0) + p(1)q(1) + p(2)q(2) for any two polynomialsp(x),q(x) E V. So, for example,

(x+2, x^2-1) = (0+2)(0^2-1) + (1+2)(1^2 - 1) + (2+2)(2^2-1) =10.

a) Show that the function(.,.) defines an inner product onV.

b) Is the set A=(1,x,x^2) an orthogonal set? (with respect tothe inner product above.)

c) Is the set B=(1,x-1,3x^2-6x+1) an orthogonal set?

d) Calculate ||1||^2, and ||3x^2-6x+1||^2

e) Is the set B an orthonormal set?

f) Write f=x^2-6x+12 as a linear combination of the vectors inB.

Please answer numbers 2c, 3a, and 3b. Please and thank you.

MAT344H1 Lecture Notes - Lecture 2: Complement Graph, Mountaineering, If And Only If

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