Abstract Algebra

  1. Putnam 2001 A1

    Consider a set \(S\) and a binary operation \(\ast\) on \(S.\) Assume \((a\ast b)\ast a=b\) for all \(a,b\in S.\) Prove that \(a\ast (b\ast a)=b\) for all \(a,b\in S.\)

  2. Putnam 2012 A2

    Let \(\ast\) be a commutative and associative binary operation on a set \(S.\) Assume that for every \(x\) and \(y\) in \(S,\) there exists \(z\) in \(S\) such that \(x\ast z= y.\) (This \(z\) may depend on \(x\) and \(y.\)) Show that if \(a,b,c\) are in \(S\) and \(a\ast c = b\ast c,\) then \(a=b.\)

  3. Putnam & Beyond

    Let \(S\) be the smallest set of rational functions containing \(f(x,y) = x\) and \(g(x,y) = y\) that is and closed under subtraction and taking reciprocals. Show that \(S\) does not contain the nonzero constant functions.

  4. Putnam & Beyond

    Assume that \(a\) and \(b\) are elements of a group with identity element \(e\) satisfying \((aba^{-1})^n = e\) for some positive integer \(n.\) Prove that \(b^n = e.\)

  5. Putnam & Beyond

    Let \(G\) be a group with no element of order \(2\) having the property that \({(ab)^2 = (ba)^2}\) for all \(a,b \in G.\) Prove that \(G\) is abelian.

  6. Putnam & Beyond

    On the set \(M = \mathbf{R}\!\setminus\!\{3\}\) define the binary operation \(\ast\) as \[x \ast y = 3(xy-3x-3y) + m,\] for \(m\in\mathbf{R}.\) What are all the possible values of \(m\) for which \(M\) is a group under the operation \(\ast?\)

  7. Putnam & Beyond

    Prove that the group of invertible \(4 \times 4\) matrices with rational entries has no elements of order seven.

  8. Putnam & Beyond

    Let \(a\) and \(b\) be elements in a ring with identity. Prove that if \(1-ab\) is invertible, then so is \(1-ba.\)

  9. MIT

    Let \(G\) be a finite group, and let \(x\) be some designated element of \(G.\) Find the number of pairs \((a,b) \in G \times G\) such that \(x = abaabab.\)

  10. Putnam 1989 B2

    Let \(S\) be a non-empty set with an associative operation that is left- and right-cancellative (\(xy = xz\) implies \(y = z,\) and \(yx = zx\) implies \(y = z\)). Assume that for every \(a\) in \(S\) the set \(\{a^n : n=1,2,3,\dotsc\}\) is finite. Must \(S\) be a group?

  11. MIT

    Let \(R\) be a ring for which \(x^2 = 0\) for all \(x \in R.\) Show that \(xyz + xyz = 0\) for all \(x,y,z \in R.\)

  12. Putnam 2012 B1

    Let \(S\) be a class of functions from \([0, \infty)\) to \([0, \infty)\) that satisfies:

    • The functions \(f_1(x) = \mathrm{e}^x - 1\) and \(f_2(x) = \ln(x+1)\) are in \(S;\)
    • If \(f(x)\) and \(g(x)\) are in \(S\), then \(f(x) + g(x)\) and \(f(g(x))\) are in \(S;\)
    • If \(f(x)\) and \(g(x)\) are in \(S\) and \(f(x) \geq g(x)\) for all \(x \geq 0,\) then the function \(f(x) - g(x)\) is in \(S.\)

    Prove that if \(f(x)\) and \(g(x)\) are in \(S\), then the function \(f(x) g(x)\) is also in \(S\).

  13. MIT

    Let \(\pi\) be a random permutation of \(1, 2,\dotsc, n.\) Fix a positive integer \(1 \lt k \lt n.\) What is the probability that in the disjoint cycle decomposition of \(\pi,\) the length of the cycle containing \(1\) is \(k?\) In other words, what is the probability that \(k\) is the least positive integer for which \(\pi^k(1)=1?\)

  14. Putnam 2007 A5

    Suppose that a finite group has exactly \(n\) elements of order \(p,\) where \(p\) is a prime. Prove that either \(n = 0\) or \(p\) divides \(n+1.\)

  15. Putnam 2010 A5

    Let \(G\) be a group, with operation \(*.\) Suppose that
    • \(G\) is a subset of \(\mathbf{R}^3\) (but \(*\) need not be related to addition of vectors);
    • For each \(\bm{a},\bm{b} \in G\), either \(\bm{a}\times \bm{b} = \bm{a}*\bm{b}\) or \(\bm{a}\times \bm{b} = 0\) (or both), where \(\times\) is the usual cross product in \(\mathbf{R}^3\).
    Prove that \(\bm{a} \times \bm{b} = 0\) for all \(\bm{a}, \bm{b} \in G.\)

  16. Putnam 1997 A4

    Let \(G\) be a group with identity \(e\) and \(\varphi\colon G \to G\) a function such that \[\varphi(g_1) \varphi(g_2) \varphi(g_3) = \varphi(h_1) \varphi(h_2) \varphi(h_3) \] whenever \(g_1g_2g_3 = e = h_1h_2h_3.\) Prove that there exists an element \(a \in G\) such that \(\psi(x) = a\varphi(x)\) is a homomorphism. I.e. \(\psi(xy) = \psi(x)\psi(y)\) \(\forall x,y \in G.\)