+1 862 207 3288 31 El be a group a-nd let 0‘ 6 0‘ {Assume “‘3‘ Owl“) “-9 p is a prime number. Prove that order of
ry cmenta .] +1,…,p-ltsa13op.(lopomts)
1 Prove that direct product of groups 22 and 25 (notation Z? x Z )is isomo hi t 1′
2w. (10 points) 5 . rp c ocyc 16 group
3 hit n. m and k be Positive integCTS. k S 7″. Assume also that k” is a multiple of m. Prove that
the function
“3) – (ks) modm, s =0,1,-a”” 1
is a homomorphism from group Zn to group 2,… Clarification: you need to verify that
f((81 ;~ 52) mod n) = (f(31) + flsz» mod m
Find the kernel of f? Justify.(10 POMS)
‘4. Let G be a group. Let 11 be the subset of G that consists of the following elements: the identity
3 and all the products of the form (tin; – – -a‘,i, where k is an integer, (ti 6 0 (products of arbitrary
length). Prove that
a) H is a normal subgroup. (Hint: show that 1:042: ” 2 (:mm“)“.) (10 points)
b) in the quotient group G / 11 every element (except for identity) either of order 4 or of order
2. (lo points)
