This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 183

2008 Gheorghe Vranceanu, 2
Tags: group theory , algebra , morphism , transposition decomposition , superior algebra
Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.

2020 IMC, 7
Tags: group theory , abstract algebra , superior algebra
Let $G$ be a group and $n \ge 2$ be an integer. Let $H_1, H_2$ be $2$ subgroups of $G$ that satisfy $$[G: H_1] = [G: H_2] = n \text{ and } [G: (H_1 \cap H_2)] = n(n-1).$$ Prove that $H_1, H_2$ are conjugate in $G.$ Official definitions: $[G:H]$ denotes the index of the subgroup of $H,$ i.e. the number of distinct left cosets $xH$ of $H$ in $G.$ The subgroups $H_1, H_2$ are conjugate if there exists $g \in G$ such that $g^{-1} H_1 g = H_2.$

1977 Miklós Schweitzer, 5
Tags: linear algebra , matrix , superior algebra
Suppose that the automorphism group of the finite undirected graph $ X\equal{}(P, E)$ is isomorphic to the quaternion group (of order $ 8$). Prove that the adjacency matrix of $ X$ has an eigenvalue of multiplicity at least $ 4$. ($ P\equal{} \{ 1,2,\ldots, n \}$ is the set of vertices of the graph $ X$. The set of edges $ E$ is a subset of the set of all unordered pairs of elements of $ P$. The group of automorphisms of $ X$ consists of those permutations of $ P$ that map edges to edges. The adjacency matrix $ M\equal{}[m_{ij}]$ is the $ n \times n$ matrix defined by $ m_{ij}\equal{}1$ if $ \{ i,j \} \in E$ and $ m_{i,j}\equal{}0$ otherwise.) [i]L. Babai[/i]

2006 Pre-Preparation Course Examination, 3
Tags: superior algebra , number theory , greatest common divisor , inequalities , abstract algebra
a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$ b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality? c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.

1979 Miklós Schweitzer, 2
Tags: superior algebra , geometry
Let $ \Gamma$ be a variety of monoids such that not all monoids of $ \Gamma$ are groups. Prove that if $ A \in \Gamma$ and $ B$ is a submonoid of $ A$, there exist monoids $ S \in \Gamma$ and $ C$ and epimorphisms $ \varphi : S \rightarrow A, \;\varphi_1 : S \rightarrow C$ such that $ ((e)\varphi_1^{\minus{}1})\varphi\equal{}B$ ($ e$ is the identity element of $ C$). [i]L. Marki[/i]

1985 Iran MO (2nd round), 6
Tags: superior algebra
In The ring $\mathbf R$, we have $\forall x \in \mathbf R : x^2=x$. Prove that in this ring [b]i)[/b] Every element is equals to its additive inverse. [b]ii)[/b] This ring has commutative property.

2012 District Olympiad, 2
Tags: quadratic , algebra , polynomial , superior algebra
Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent: (a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$. (b) $(A,+,\cdot)$ is a field.

1967 Miklós Schweitzer, 3
Tags: superior algebra , group theory , abstract algebra
Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup. [i]B. Csakany[/i]