Tour:Hard nuts four (beginners)

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Additive cyclic groups

1. NEEDS LOT OF THOUGHT: Suppose $H,K,L$ are subgroups of $\mathbb{Z}$. Prove (by reducing the problem to one involving lcm and gcd, and then using the definitions of lcm and gcd in terms of prime factorization) that:
   • $⟨H,K\cap L⟩=⟨H,K⟩\cap ⟨H,L⟩$
   • $H\cap ⟨K,L⟩=⟨H\cap K,H\cap L⟩$
2. NEEDS SOME THOUGHT: Prove the results above results, replacing $\mathbb{Z}$ by any cyclic group.
3. NEEDS SOME THOUGHT: Let $A$ be a finite subset of $\mathbb{Q}$. Prove that the subgroup generated by $A$ is cyclic. (A group with the property that every finite subset generates a cyclic group is termed a locally cyclic group).
4. Suppose $(G,+)$ is an Abelian group and $g\in G$, such that there exist $a,b\in G$, such that $2a=3b=g$. Prove that there exists a $c\in G$ such that $6c=g$. Generalize the result replacing 2,3 by any two integers, and replacing 6 by the least common multiple of these integers.
5. Using the previous problem, prove that if an arithmetic progression of integers contains a perfect square and a perfect cube, then it contains a perfect sixth power. (Hint: First use the previous problem to solve the case where the common difference is relatively prime to the terms of the progression)
6. Prove that the multiplicative group of all roots of unity in $\mathbb{C}$, is locally cyclic: the subgroup generated by any finite subset is cyclic.
7. NEEDS LOT OF THOUGHT: Prove that any additive subgroup of $\mathbb{R}$ is either dense or infinite cyclic, generated by the positive element of smallest magnitude in it. (Dense here means that it intersects every open interval, and cyclic means that it is additively generated by an element). This is a generalization of the fact that any subgroup of $\mathbb{Z}$ is infinite cyclic.

Multiplicative groups

For these exercises, use the fact that if $p$ is prime, the multiplicative group mod $p$ is a cyclic group. Direct application of Fermat's little theorem, and the concept of order of an element, are useful.

1. An element $a$ mod $p$ (not zero mod $p$) is termed a quadratic residue mod $p$ if there exists $x$ such that $x^2\equiv amodp$. Otherwise $a$ is termed a quadratic nonresidue. Prove that for an odd prime $p$, there are exactly $(p-1)/2$ quadratic residues and $(p-1)/2$ quadratic nonresidues. Further, show that $a$ is a quadratic residue mod $p$ if and only $a^{(p-1)/2}\equiv 1modp$, and $a$ is a quadratic nonresiude if and only if $a^{(p-1)/2}\equiv -1modp$.
2. Prove that the quadratic residues form a subgroup of index two, and the quadratic nonresidues form the other coset.
3. A primitive element mod $p$ is an element that generates the multiplicative group mod $p$. Prove that there are exactly $\phi (p-1)$ primitive elements.
4. The $n^{th}$ Fermat number is defined as: $F_n=2^{2^n}+1$. Prove that if $p$ is a prime divisor of $F_n$, then $p-1$ is a multiple of $2^{n+1}$.
5. An element $a$ mod $p$ (not zero mod $p$) is termed a cubic residue mod $p$ if there exists $x$ such that $x^3\equiv amodp$. Prove that the cubic residues form a multiplicative subgroup of the multiplicative group mod $p$. Further prove that if $3|p-1$, then this subgroup has index three, and otherwise, the subgroup is the whole group.
6. Prove that if $q$ is a prime divisor of $a^p-1$ for a prime $p$ and an integer $a>1$, then either $p|q-1$ or $q|a-1$.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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