1. Use Lagrange's Theorem to determine the possible subgroup sizes in a group with exactly 40...
1. Both Lagrange's theorem and Cauchy's theorem deal with the relationship between the size of a group and the order of its elements. (a) Explain the difference between the theorems in general terms and by using S7 as an example. Your explanation should include what we can and cannot conclude from each theorem about S7 (b) Which theorem would allow you to prove that if a group contained only elements that had order some power of 2, then the order...
12pts) 1. Both Lagrange's theorem and Cauchy's theorem deal with the relationship between the size of a group and the order of its elements. (a) Explain the difference between the theorems in general terms and by using S, as an example. Your explanation should include what we can and cannot conclude from each theorem about S7. (b) Which theorem would allow you to prove that if a group contained only elements that had order some power of 2, then the...
This is 2(b): The following exercise shows that the converse to Lagrange's theorem is false, i.e. even if d ||G|, there need not be a subgroup of G with order d. (a) Let n > 4 and consider the alternating group An. Suppose that NC An is a normal subgroup and that there is a 3-cycle (abc) E N. Prove that N = An. Hint: it is enough to show that N contains all 3-cycles. What is the conjugate of...
I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
2. Use Lagrange's theorem to prove the Euler-Fermat Theorem: If n E Z+ and (a, n) = 1, then ap(n)-1 mod n.
I have to use the following theorems to determine whether or not it is possible for the given orders to be simple. Theorem 1: |G|=1 or prime, then it is simple. Theorem 2: If |G| = (2 times an odd integer), the G is not simple. Theorem 3: n is an element of positive integers, n is not prime, p is prime, and p|n. If 1 is the only divisor of n that is congruent to 1 (mod p) then...
1-5 theorem, state it. Define all terms, e.g., a cyclic group is generated by a single use a element. T encourage you to work together. If you find any errors, correct them and work the problem 1. Let G be the group of nonzero complex numbers under multiplication and let H-(x e G 1. (Recall that la + bil-b.) Give a geometric description of the cosets of H. Suppose K is a proper subgroup of H is a proper subgroup...
please show step by step solution with a clear explanation! 2. Let G be a group of order 21. Use Lagrange's Theorem or its consequences discussed in class to solve the following problems: (a) List all the possible orders of subgroups of G. (Don't forget the trivial subgroups.) (b) Show that every proper subgroup of G is cyclic. (c) List all the possible orders of elements of G? (Don't forget the identity element.) (d) Assume that G is abelian, so...
#11 11. If a group G has exactly one subgroup H of order k, prove that H is normal in G. Define the centralizer of an element g in a group G to be the set 12.
Consider the rectangle shown, and let A be the eight points listed. The symmetry group, G, of this rectangle has four elements: the identity j a flip over a horizontal axis through its centre v a flip over a vertical axis through its centre r a rotation about its centre by 180 degrees. We regard G as a subgroup of SA Page 2 (a) Find G(a) and G(b), the orbits of a and b. (b) Find Ga and Gb, the...