Abelian group

If G is any group, define $:G->G by $(g) = g^-1. Show that G is abelian if an only if $ is a homomorphism.

Homomorphisms

If $:G->G1 is a homomorphism, show that K = the set of g belonging to G given that $(g)=1 is a subgroup of G (called the kernel of $)

Isomorphism

Note: ~~ means an isomorphism exists. Moreover,if an isomorphism existed from G to G1 I would say G ~~ G1 Questions: If G is an infinite cyclic group, show that G ~~ Z (Z is the set of integers)

Homomorphisms

If G = and $:G->G1 is an onto homomorphism, show that G1 = <$(X)>, where $(X) = the set of $(x) given that x belongs to X.

Cosets

If H is a subgroup of G, define a mapping $ from the right cosets of H to the left cosets by $(Ha) = a^-1H. Show that $ is a (well defined) bijection.

Cosets/subgroups

Let G = RxR (R is the real numbers) with addition (x,y) + (x', y') = (x+x', y+y'). Let H be the line y=mx through the origin: H = {(x,mx)such that x belongs to R (R is real numbers). Show that H is a subgroup of G and describe the cosets H + (a,b) geometrically.

Subgroups

"C" means set containment (not proper set containment) and "T" means intersection of sets If H and K are subgroups of a group and |H| is prime, show that H C K or H T K = {1}

Groups and elements

If G is a group of order p^k, where p is a prime and k >=, show that G must have an element of order p.

Cosets

note: C means set containment (not proper) |G:H| means index of subgroup H in G U means union of sets E means belonging to Let K C H C G be groups. Show that both |G:H| and |H:K| are finite if and only if |G:K| is finite, and then |G:K| = |G:H||H:K|. Hint: if |H:K| = n, let Kh1, Kh2, ..., Khn be the distinct cosets of ...continues

Lagrange's theorem

Show that A4 has no subgroup of order 6 and hence that the converse of Lagrange's theorem is false. Note: "An" is the alternating group of degree n