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Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866.

My knowledge of group theory is undergraduate-level stuff. I'm looking at the paper cited above, to which I was referred in this answer.

After a prefatory section, this paper says

"We begin by recalling"[ . . . ]

The word recalling suggests this is standard stuff within some community.

the fundamental correspondence between the representations of $G$ and the subsets of the subgroup lattice of $G$:

Apparently "representations" in this context does not mean the group operation will correspond to matrix multiplication or to composition of linear transformations, but rather that it will correspond to composition of permutations of finite sets.

In the first sentence the paper, Johnson refers to "the least value of the positive integer $n$ such that $G$ can be embedded in the symmetric group of degree $n$", and I wonder if that's what he means by "$n$" below:

$$ \begin{array}{lcl} \text{representation $\rho$} & \phantom{mmm} & \{G_1,\cdots,G_n\} \\ \text{degree of $G$} & & \sum_{i=1}^n |G:G_i| \\ \text{number of transitive} \\ {}\quad\text{constituents of $\rho$} & & n \\ \text{$\rho$ transitive} & & n=1 \end{array} $$

. . . and it goes on from there.

So I surmise that "transitive constituents" means two members of the set being permuted belong to the same "transitive constituent" of $\rho$ iff some member of $G$ moves one of them to the other. That's obviously an equivalence relation. So

  • Is that surmise right?
  • Is my guess that $n$ means the same thing here that it meant earlier right?
  • What do the first two lines of the table mean? I'm guessing $G_1,\ldots,G_n$ are subgroups, so how does specifying a sequence of subgroups specify a representation? I'm also guessing $|G:G_i|$ is the order of a quotient group, but here I hesitate since it didn't say $|G/G_i|$.
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Michael Hardy
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  • Are you familiar with the permutation action of a group $G$ on the set of cosets of a subgroup $G_i$ of $G$? The number of such cosets is equal to the index of $G_i$ in $G$, which is denoted by $|G:G_i|$. (This is not a quotient group.) The paper is based on that construction, so you need to understand that before attempting to understand the paper. – Derek Holt Oct 28 '13 at 07:51
  • @DerekHolt : Do you have in mind simply that if $g\in G$ then $gG_i={gh : h\in G_i}$? ${}\qquad{}$ – Michael Hardy Oct 28 '13 at 17:32
  • Yes that's right. If you let $X_i$ be the set of $|G|/|G_i|$ distinct left cosets of $G_i$ in $G$, then there is a homomorphism $f_i$ from $G$ to the symmmetric group $S_{X_i}$ on $X_i$, where $f_i(x)$ acts on $X_i$ by mapping $gG_i$ to $xgG_i$. The representations being considered by Johnson are defined by combining several of these $f_i$ together to get a homomorphism $F:G \to S_X$ with $X = \cup_{i=1}^k X_i$ for some $k$. – Derek Holt Oct 28 '13 at 19:19
  • @DerekHolt : OK, I'm working on it, but other things interrupt me. Any chance I can convince you to write $\bigcup_{i=1}^k X_i$ instead of $\cup_{i=1}^k X_i$? Generally \cup is used for things like $X_1\cup X_2\cup\cdots\cup X_k$ and \bigcup is used for things like $\bigcup_{i=1}^k X_i$. And when it's in a "diplayed", rather than "inline", setting, it looks like this: $\displaystyle\bigcup_{i=1}^k X_i$. ${}\qquad{}$ – Michael Hardy Oct 30 '13 at 02:02
  • @DerekHolt and others. The point Derek Holt made above is clear once explained, and once that's explained, it's routine to show that every permutation representation is isomorphic to one of those described in this way by the notation "${G_1,\ldots,G_n}$". The paper goes on to say other things about these that presuppose prior familiarity with some conventions, including a convention according to which it matter in what order $G_1,\ldots,G_n$ are listed here. So my next question is: What is the best book in which to learn standard things like these? – Michael Hardy Nov 03 '13 at 17:42

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