abstract algebra

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.

abstract algebra

"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} Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

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. Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

note: C means set conatainment (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 K in H. Show that Hg = Kh1g UKh2g U ......U Khng is a disjoint union for all g E G Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

Show that A4 has no subgroup of order 6 and hence that the converse of Lagrange's theorem is false. Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks. Note: "An" is the alternating group of degree n