Let $G$ be a finite group. Can one always construct a faithful representation of $G$ over $\mathbb C$ of dimension $\le \sqrt{|G|}$? What would be the order of the minimal such dimension for a "random" group? Are the worst groups (with the largest minimal faithful dimensions) special in some way?
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6$\begingroup$ The smallest dimension of a faithful complex representation of $C_2 \times C_2 \times C_2$ is $3$, which is a little larger than $\sqrt{8}$. (But that's the only counterexample I can think of.) $\endgroup$– Derek HoltCommented Feb 4, 2020 at 22:00
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3$\begingroup$ There is a result, proven using the classification of finite simple groups, which says that if $G$ is finite and not cyclic of prime order, then $G$ contains a proper subgroup $H$ with $|H| \geq \sqrt{|G|}$. Then if such a $H$ has trivial core (ie $H$ does not contain any nontrivial normal subgroups of $G$), the permutation representation on cosets of $H$ is faithful of degree $[G:H] \leq \sqrt{|G|}$. $\endgroup$– spinCommented Feb 4, 2020 at 22:43
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2$\begingroup$ If $G$ is simple, then faithful reps are the same as non-trivial reps and the minimum dimension of a complex representation is known thanks to Landazuri-Seitz on the Lie type case (modulo the odd correction here and there). The best list I am aware of is in a paper by Guest, Morris, Praeger and Spiga (email me if you need a copy). Running through that list, the largest rep (as a power of $|G|$) occurs in the family $PSL_2(q)$ where it is roughly $|G|^{1/3}$. $\endgroup$– Nick GillCommented Feb 5, 2020 at 11:22
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1$\begingroup$ It's clear that if $G$ is a finite group with no faithful complex representation of degree at most $\sqrt{G}$, and $\theta$ is a faithful complex character of $G$ of least degree, then $\theta$ is the sum of $t \geq 2$ distinct non-trivial irreducible characters, say $\chi_{1}, \chi_{2}, \ldots, \chi_{t}$, no two of which are algebraically conjugate. Also, if we set $K_{i} = \cap_{j \neq i }{\rm ker}\chi_{j}$, then $1 \neq K_{i} \neq G$ for each $i$, and the product $K_{1} K_{2} \ldots K_{t}$ is direct. $\endgroup$– Geoff RobinsonCommented Feb 5, 2020 at 16:27
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3$\begingroup$ Note that if $G$ is an extraspecial $2$-group of order $2^{2n+1}$, then the smallest degree of a faithful complex character of $G$ is $2^{n} = \sqrt{\frac{|G|}{2}}$. $\endgroup$– Geoff RobinsonCommented Feb 5, 2020 at 17:19
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I'm writing a paper where I prove that all counterexamples are related to Derek Holt's example: if G is a finite group then either G has a faithful representation over $\mathbb{C}$ of dimension $\leq\sqrt{|G|}$ or $G$ is a $2$-group with center is elementary abelian of order 8 and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ vanish on $G-Z(G)$.
For any of these $2$-groups, the minimal dimension of a faithful representation is $\frac{3}{\sqrt{8}}\sqrt{|G|}$.
I also prove that this minimal dimension is equal to $\sqrt{|G|}$ if and only if $G$ is a $2$-group with center elementary abelian of order either $4$ or $16$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ vanish on $G-Z(G)$.
I'd be happy to share my preprint. It would help me to know the motivation for this question.
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1$\begingroup$ It's maybe more an analogy than a motivation (and I'm not the original poster), but Ado's theorem says that every complex finite-dimensional Lie algebra has a faithful finite-dim rep, but the known upper bounds are extremely high, even for nilpotent Lie algebras: exponential in the dimension if I remember correctly. It's not even known if the best upper bound grows linearly. $\endgroup$– YCorCommented Jan 27, 2021 at 12:46
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$\begingroup$ I hope you share here the link to arXiv (or any other repository) when available. $\endgroup$– YCorCommented Jan 27, 2021 at 12:47
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$\begingroup$ @Ycor, thanks for your comments. Sure, I'll upload it to arXiv and share the link here when ready. $\endgroup$ Commented Jan 27, 2021 at 14:13
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$\begingroup$ Dear Alexander, thanks for your post. The motivation, to be honest, was pure curiosity, but I like the result! As Yves says, please do share here the link to the preprint once it is on arxiv. Maybe someone else will comment. $\endgroup$ Commented Jan 28, 2021 at 9:07
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3$\begingroup$ Here it the link to my preprint: arxiv.org/abs/2102.01463 $\endgroup$ Commented Feb 3, 2021 at 8:23