Recent content by mathmajor2013

  1. M

    Graduate Prove Prime Ideal Problem: I/J ⊆ P

    If i like ab in I intersect J, then ab is in P. Therefore a in P or b in P since P is prime. Neither a or b need be in I intersect J though.
  2. M

    Graduate What Determines Equality of Principal Ideals in an Integral Domain?

    Let R be an integral domain with elements a,b in R and <a>,<b> the corresponding principal ideals. Prove that <a>=<b> if, and only if, a=bu for some unit u in U(R). proof if a=bu, then ar=bur. Since ur in R, call it s. So ar=bs for some s in R. Therefore a times some elements in r is equal...
  3. M

    Graduate Prove Prime Ideal Problem: I/J ⊆ P

    Let R be a ring with ideals I, J, and P. Prove that if P is a prime ideal and I intersect J is a subset of P, then I is a subset of P or J is a subset of P.
  4. M

    Undergrad True or False? Continuity Problem: f(0)=g(0)

    1. Homework Statement True or false? If f and g are continuous at 0 and f(1/(2n+7))=g(1/(7-2n)) for all positive integers n, then f(0)=g(0). 2. Homework Equations lim x->0 f(x)=f(0) lim x->0 g(x)=g(0) 3. The Attempt at a Solution NO CLUE. My intuition says false.
  5. M

    Can Continuity at 0 Guarantee Equality at 0?

    Homework Statement True or false? If f and g are continuous at 0 and f(1/(2n+7))=g(1/(7-2n)) for all positive integers n, then f(0)=g(0). Homework Equations lim x->0 f(x)=f(0) lim x->0 g(x)=g(0) The Attempt at a Solution NO CLUE. My intuition says false.
  6. M

    ANALYSIS: Prove that lim x->c of sqrt{f(x)} = sqrt{L}

    Homework Statement Suppose that f(x)>=0 in some deleted neighborhood of c, and that lim x->c f(x)=L. Prove that lim x->c sqrt{f(x)}=sqrt{L} under the assumption that L>0. Homework Equations When 0<|x-c|<delta, |f(x)-L|<epsilon. The Attempt at a Solution When, 0<|x-c|<delta...
  7. M

    Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

    Oh I see. Yes! Thank you.
  8. M

    Is f a Surjective and Injective Isomorphism from HxN to HN in G?

    Right the homomorphism part is easy now. Am I able to use the pigeonhole principle for the isomorphic part? That is, are HxN and HN the same size? It seems like they are since H intersect N is only the identity.
  9. M

    Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

    I thought that G/H was the set of left or right cosets, not a coset itself? But yes that does help, thank you!
  10. M

    Is f a Surjective and Injective Isomorphism from HxN to HN in G?

    I am confused how to start this problem. To first show it is a homomorphism, is f((h,n)(h',n'))=f((hh',nn'))?
  11. M

    Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

    I'm lost on this one. It doesn't make sense how the number of left cosets corresponds to the normality. #gH=#Hg doesn't seem like it necessarily means that gH=Hg.
  12. M

    Is f a Surjective and Injective Isomorphism from HxN to HN in G?

    Let G be a group, H a normal subgroup, N a normal subgroup, and H intersect N = {e}. Let H x N be the direct product of H and N. Prove that f: HxN->G given by f((h,n))=hn is an isomorphism from HxN to the subgroup HN of G. Hint: For all h in H and n in N, hn=nh.
  13. M

    Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

    Let G be a group and H be a subgroup of G. Prove that if [G:H]=2, then H is a normal subgroup of G.
  14. M

    Graduate |G|=p^k. Prove G has an element of order p.

    The pth root of x^m? I'm sorry I cannot see where this one is going
  15. M

    Graduate |G|=p^k. Prove G has an element of order p.

    This kind of seems like a contradiction because m/p is smaller than m, yet x^m/p=(x^m)^1/p=e^1/p=e.