Suppose that G is a group of order 80.
(a) Show that G is not simple.
(b) How that G is solvable.
Suppose that G is a group of order 80. (a) Show that G is not simple....
4. Suppose G is a group of order n < 0. Show that if G contains a group element of order n, then G is cyclic.
Show that if G is a group of order np where p is prime and 1 <n<p, then G is not simple.
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
Use Sylow's theorem to show that group of order 256 is not simple.
(i) State Sylow's theorems. (ii) Suppose G is a group with IGI pr where p, q and r are distinct primes. Let np, nq and nr, denote, respectively, the number of Sylow p, q- and r-subgroups of G. Show that Hence prove that G is not a simple group. (iii) Prove that a group of order 980 cannot be a simple group.
Let G be a finite group, and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G. [consider the subgroup
of G]
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(4) Simplicity II. In this problem, you show that a group G with |G= 30 is not simple. can G have? How many Sylow 3-subgroups? Sylow p-subgroups of G, then either P (a) How many Sylow 5-subgroups (b) Show that, if P and P Pn P(e} (hint: this fact is not true in general, so think again if you didn't use something special to G). (c) If G has the largest possible number of Sylow 5-subgroups, then how many elements...
4. Suppose G is a finite simple group with a subgroup H such that G: H = n. Prove that there is an injective homomorphism 0:G + Sn. (Added: assume n > 1). (4 marks)
14. Suppose that G is a group in which {1} and G are the only subgroups. Show that G is finite and, in fact, is cyclic of order 1 or a prime.
Let G be a group of order 35. Show that every non-trivial subgroup of G is cyclic.