Prove that a group of order 56 has a normal Sylow p-subgroup for some prime p dividing its order

Prove that a group of order 56 has a normal Sylow p-subgroup for some prime p dividing its order

Irreducible Polynomial : Galois Group and Splitting Field

4. Find an irreducible polynomial defining the field extension K = Q (cube root 2, sq root − 3) over Q . Is K a normal extension of Q ? What is the Galois group for the splitting field of the polynomial defining K over Q ?

Describe the Nonabelian Group

Let G be a finite nonabelian group of order 27 where all the elements have order 3. Prove that there is exactly one such group G and give a complete description.

Subset and Subgroup Problem

If G is a group and gEG, define C(g)={zEG|zg=gz} show that C(g) is a subgroup of G (the centralizer of g in G). Note: E means element of

Cyclic Group Problem

Let G be a cyclic group of order n. Show that g^n=1 for all gEG. If g^m=1 in G where gcd(m,n)=1, show that g=1

Wilson's Theorem : Cyclic Groups and Order of an Element

13.a) If G={g1,g1,....,gr} is an abelian group, show that g1,g2....gr equals the product of the elements of order 2. b) Prove Wilson's Theorem: If p is a prime then (p-1)! R (-1)(modp) note: R is a equivalence relation

Groups and Integers

Let m be the smallest positive integer such that @^m=E for all @eS_n. Show that m=lcm(2,3,4,5,...,n). note: e denotes element of

Subgroups

If X is a nonempty subset of a group G, let ={x1^(k1),x2^(k2)...xm^(km)|m>=1, xiEX and kiEZ for each i}. a) show that is a subgroup of G that contains x. b) show that C=H for every subgroup H such that XC=H. Thus is the smallest subgroup of G that contains X, and is called the subgroup generated by X. note: ...continues

Subgroups

If K is a subgroup of H and H is a subgroup of G, is K a subgroup of G? Please justify your answer.

Cyclic Groups

Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG)