query: If an $H\leq G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of atleast dimension $d$.
positive_passages.docid: 143095
positive_passages.text: If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.
negative_passages.docid: 62077
negative_passages.text: Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
negative_passages.docid: 24626
negative_passages.text: Irreducibility of $X^n - 2$ over $\mathbf Z[i]$.
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
negative_passages.docid: 65162
negative_passages.text: Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.
negative_passages.docid: 10998
negative_passages.text: Embedding $S_n$ in $A_{2n}$
negative_passages.docid: 135808
negative_passages.text: Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
negative_passages.docid: 70249
negative_passages.text: Irreducibility of Polynomials in $k[x,y]$
negative_passages.docid: 94469
negative_passages.text: How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
negative_passages.docid: 22477
negative_passages.text: If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
negative_passages.docid: 127310
negative_passages.text: Suppose $\gcd(\deg(f),\deg (g))=1$. Show that $g(x)$ is irreducible in $k(\alpha)[X]$.
negative_passages.docid: 151621
negative_passages.text: $G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
negative_passages.docid: 121587
negative_passages.text: Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
negative_passages.docid: 10627
negative_passages.text: When is $X^n-a$ is irreducible over F?
negative_passages.docid: 34724
negative_passages.text: For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?