5. If gcd(m, n) = d, then there are d homomorphisms from Zm to Zn. Identify the four homomorphisms from Z12 to Z20

Find all monic irreducible polynomials of degree 2 over Z3.

6. Prove that if is a permutation on , then is a permutation on .

Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]

1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.

2. List all the elements of (Z30, Đ) that are of order 15.

Negative points per question: Based on Section 4.2 of HMU. 1. h is a homomorphism from the alphabet {a,b,c} to {0,1}. If h(a) = 01, h(b) = 0, and h(c) = 10, which of the following strings is in h¯¹ (010010)? a) babc b) bacb c) cbbc d) abbc 2. The language of regular expression (0+10)* is the set of all strings of O's and 1's such that every 1 is immediately followed by a 0. Describe the complement of this language (with respect to the alphabet {0,1}) and identify in the list below the regular expression whose language is the complement of L((0+10)*). a) (0+1)*11(0+1)* + (0+10)*1 b) (0+10)*11(0+1)* c) (1+01)* d) (0+10)*1(ɛ+11(0+1)*) 3. If h is the homomorphism defined by h(a) = 0 and h(b) = ε, which of the following strings is in h¨¹ (000)? a) abaabaa b) abbbabaab c) abaabbb d) babab 4. Find, in the list below, a regular expression whose language is the reversal of the language of this regular expression: 10(2+3). Recall that the reversal of a language is formed by reversing all its…

Show that Z3 X Z5 is cyclic.

3. Are the rings 2Z, 3Z isomorphic. Any isomorphism is linear for addi- tion, so it has the form f(x) = kx. Is there some k which commutes with multiplication? Another approach is to ask if the second ring has a nonzero element y such that y? = 2y. 6. Show tho ring Sa Iba/2lx b 7Uhag ovactly 2 9utomornhisms Thoro

Is ZXZ cyclic? [Here Z means (Z, +).] Shon