group theory Example 6.7 Show that the group G((a,b a",b,aba b)) (pand q are relatively prime) is isomorphic to...
group theory
Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and q are relatively prime. Solution arch hp
Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and q are relatively prime. Solution arch hp
group theory
Example 10.3 Show T trasformation from presentation x |x° ) to the presentation (a,bla,b,aba b where p and q are relatively prime. ENG A INTL DELL
Example 10.3 Show T trasformation from presentation x |x° ) to the presentation (a,bla,b,aba b where p and q are relatively prime. ENG A INTL DELL
(h) Show that the affine group AGL(1,q) is isomorphic to a subgroup of GL(2,9), the general linear group of non-singular matrices over GF(q), by using the mapping ax + b (Why is this an isomorphism?) [10] (8 h
group theory
2. Consider the group presentation (a,b a1,b3,aba b Determine a van Kampen lemma word for W ab ab dint Inr a. 0
2. Consider the group presentation (a,b a1,b3,aba b Determine a van Kampen lemma word for W ab ab dint Inr a. 0
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Prove that if G is a group and a, b ∈ G with aba^(−1) = b^j , then a^r ba^−r = b^(j)^(r) (Hint: recall that ab^(t)a^ (−1) = (aba^−1 )^t ).
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
Suppose a and b are numbers that are relatively prime to p. Show that at least one of the three numbers, a, b or ab, must be a quadratic residue.
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If m and n are any positive integers, again relatively prime
Show that if G is a group of order np where p is prime and 1 <n<p, then G is not simple.