MAT-3110 Midterm: MATH 3110 App State Fall2015 Test1

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turquoisegnu128

15 Feb 2019

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Be sure to show your work: (20 points) de nition and basics (a) suppose that g is a non-empty set equipped an operation. 5: (b) let g = z>0 be the set of positive integers. De ne a binary operation as follows: x y = xy (example: G paired with this operation is not a group. [use concrete counterexample(s). ] (c) let g = (cid:8)n 2 | n q6=0(cid:9). (the non-zero rational multiples of 2. For example: 2 and belong to g. ) g is not a group under multiplication. If so, why so and what is its inverse? (d) what is the order of 80 in z100? (e) draw the subgroup lattice for z100: (25 points) more modular arithmetic. (a) list the elements of u (10). Then nd their orders and the list the elements in cyclic subgroup generated by that element. [note: there may be more spaces than you need. ] x =

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These questions are asking about subrgoups, groups, isormorphic.

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(below question)

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.

ey table is seen below xeecions tor Chapter 1 12, Fori, b â¬ Î.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

can some help me with number 2 number 1 is needed to solve to get 2 please and thank you .

MAT-3110 Midterm: MATH 3110 App State Fall2015 Test1

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