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(10 pts) Let G be a finite group acting on a set X. Prove that the he number of orbits equals the...
(10) Let G be a finite group. Prove that if H is a proper subgroup of...
(10) Let G be a finite group. Prove that if H is a proper subgroup of G, then |H| = |G|/2. (11) Let G be a group. Prove that if Hį and H2 are subgroups of G such that G= H1 U H2, then either H1 = G or H2 = G.
Problem 1. Let G be a finite group and f : G → G a group...
Problem 1. Let G be a finite group and f : G → G a group automorphism ( isomorphism for G to G) of order 2 (i.e. f(f(x)) = x), and f has no nontrivial fixed points (i.e. f(x) = x if and only if x = 1). Prove that G is an abelian group of odd order.
Let a ∈ G where G is a group. If X ⊆ G is a finite...
Let a ∈ G where G is a group. If X ⊆ G is a finite subset, write Xa = {xa | x ∈ X}. Show that X and Xa have the same number of elements.
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group actio...
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group action is properly discontinuous and free prove that the quotient space M/G is a manifold. For this problem properly discontinuous means that if K c M is compact then the set {ge G | g(K) n/Kメ0) is finite) and free means the only element of g that fixes any point of M is the identity.
3. Let M be...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
Automata: solve a - e 2. (10+10+10+10+10-50 points) Agrammar is a 4-tuple G, G-ON,E,11,L$) where N is a finite set of nonterminal symbols Σ is a finite set of terminal symbols is a finite set of...
Automata: solve a - e
2. (10+10+10+10+10-50 points) Agrammar is a 4-tuple G, G-ON,E,11,L$) where N is a finite set of nonterminal symbols Σ is a finite set of terminal symbols is a finite set of rules S is the starting symbol Let N- (S, T s-{a, b, c} s-> ab aT >aaTb aT-ac S is the starting symbol. (a 10 points) Prove that the given grammar G is a context sensitive grammar. (b-10 points) What is the language L-...
(9) Let G be a group, and let x E G have finite order n. Let...
(9) Let G be a group, and let x E G have finite order n. Let k and l be integers. Prove that xk = xl if and only if n divides l_ k.
Let G be a group and g E G such that g) is finite. Let og...
Let G be a group and g E G such that g) is finite. Let og be the automorphism of G given by 09(x) = grg- (a) Prove that $, divides g. (b) Find an element b from a group for which 1 < 0) < 1b.
Let G be a finite group, prove that for every a in G there exists a positive integer n such that ...
let G be a finite group, prove that for every a in G there exists a positive integer n such that an=e, the identity.
Let X be a finite set on which G acts, let ρ be the corresponding permutation representation [cf. 1.2, example (c)], an...
Let X be a finite set on which G acts, let ρ be the corresponding permutation representation [cf. 1.2, example (c)], and xx be the character of p. Let s E G; show that xx(s) is the number of elements of X fixed by s.
Let X be a finite set on which G acts, let ρ be the corresponding permutation representation [cf. 1.2, example (c)], and xx be the character of p. Let s E G; show that xx(s)...
(10) Let G be a finite group. Prove that if H is a proper subgroup of G, then |H| = |G|/2. (11) Let G be a group. Prove that if Hį and H2 are subgroups of G such that G= H1 U H2, then either H1 = G or H2 = G.
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group action is properly discontinuous and free prove that the quotient space M/G is a manifold. For this problem properly discontinuous means that if K c M is compact then the set {ge G | g(K) n/Kメ0) is finite) and free means the only element of g that fixes any point of M is the identity.
3. Let M be...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
Automata: solve a - e
2. (10+10+10+10+10-50 points) Agrammar is a 4-tuple G, G-ON,E,11,L$) where N is a finite set of nonterminal symbols Σ is a finite set of terminal symbols is a finite set of rules S is the starting symbol Let N- (S, T s-{a, b, c} s-> ab aT >aaTb aT-ac S is the starting symbol. (a 10 points) Prove that the given grammar G is a context sensitive grammar. (b-10 points) What is the language L-...
(9) Let G be a group, and let x E G have finite order n. Let k and l be integers. Prove that xk = xl if and only if n divides l_ k.
Let G be a group and g E G such that g) is finite. Let og be the automorphism of G given by 09(x) = grg- (a) Prove that $, divides g. (b) Find an element b from a group for which 1 < 0) < 1b.
Let X be a finite set on which G acts, let ρ be the corresponding permutation representation [cf. 1.2, example (c)], and xx be the character of p. Let s E G; show that xx(s) is the number of elements of X fixed by s.
Let X be a finite set on which G acts, let ρ be the corresponding permutation representation [cf. 1.2, example (c)], and xx be the character of p. Let s E G; show that xx(s)...
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