Let G be a group of order p
am where p is a prime not dividing m. Show the following
1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ∅.
2. If P ∈ Sylp(G) and Q is any p-subgroup of G, then there exists g
∈ G such that Q 6
gP g−1
; i.e. Q is contained in some conjugate of P. In particular, any
two Sylow p-
subgroups of G are conjugate in G.
3. np ≡...
11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p,
[2]p, . . . , [p-1]p}. Prove that for y ≠ 0,
Ly restricts to a bijective map
Ly|s : S → S.
11.5 Prove Fermat's Little Theorem
Let p be an odd prime (i.e., any prime other than 2). Form two
vector spaces V1, V2 over Fp
(prime field of order p) with bases corresponding to the edges and
faces of an icosahedron (so that V1 has dimension 30 and
V2 has dimension 20). Let
T : V1 → V2 be the linear transformation
defined as follows: given a vector v ∈ V1, T(v) is the
vector in V2 whose component corresponding to a given
face is...
(a) Let G be a finite abelian group and p prime with p | | G |.
Show that there is only one p - Sylow subgroup of G. b) Find all p
- Sylow subgroups of (Z2500, +)
Let gcd(a, p) = 1 with p a prime. Show that if a has at least
one square root, then a has exactly 2 roots. [hint: look at
generators or use x^2 = y^2 (mod p) and use the fact that ab = 0
(mod p) the one of a or b must be 0(why?) ]
Throughout this question, let G be a finite group, let p be a
prime, and suppose that H ≤ G is such that [G : H] = p.
Let G act on the set of left cosets of H in G by left
multiplication (i.e., g · aH = (ga)H). Let K be the set of elements
of G that fix every coset under this action; that is,
K = {g ∈ G : (∀a ∈ G) g · aH...
let E be a finite extension of a field F of prime
characteristic p, and let K = F(Ep)
be the subfield of E obtained from F by adjoining the pth powers of
all elements of
E. Show that F(Ep) consists of all finite linear combinations of
elements in Ep with
coefficients in F.
Let p be an integer other than 0, ±1.
(a) Prove that p is prime if and only if it has the property
that whenever r and s are integers such that p = rs, then either r
= ±1 or s = ±1.
(b) Prove that p is prime if and only if it has the property
that whenever b and c are integers such that p | bc, then either p
| b or p | c.
Let p be a prime and d a divisor of p-1. show that the d th
powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this
subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6