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From Gallian, Contemporary Abstract Algebra:

...if G is a group and H is a subgroup of G, it is not always true that aH = Ha for all a in G. There are certain situations where this does hold, however, and these cases turn out to be of critical importance in the theory of groups. It was Galois, about 185 years ago, who first recognized that such subgroups were worthy of special attention.

A subgroup H of a group G is called a normal subgroup of G if aH = Ha for all a in G. We denote this by H $\triangleleft$ G.

Are subgroups with aH = Ha named normal ("usual, typical, or expected"), from they being excess in examples, and not being rare? Or is there any other reason? And is there any reason for choosing the notation $\triangleleft$?

I am asking these questions, as knowing the relation with names and notation, seems to eliminate keeping memory for the mapping of name and to what it corresponds to, via relation.

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  • $\begingroup$ Related: Triangle, right angles?; $\endgroup$ – Immortal Player Jan 24 at 10:26
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    $\begingroup$ The word "normal" is famous within math for its range of different unrelated meanings: normal subgroups, normal operators, normal distribution, normal family (of functions), etc. In group theory, normal subgroups are definitely nice but for general finite groups they are not "in excess" (look up simple groups, another peculiar label to the uninitiated). Normal subgroups used to be called "invariant" subgroups (that is, invariant under conjugation), which had a good justification. I think the usage of "normal" in group theory simply must be learned. $\endgroup$ – KCd Jan 24 at 12:33
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    $\begingroup$ See the answer by Conifold to hsm.stackexchange.com/questions/7764/… for one of the reasons "normal" is used in math, but it does not directly (it seems) apply to normal subgroups. $\endgroup$ – KCd Jan 24 at 12:39
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    $\begingroup$ Named after the famous mathematician "Abbie Normal" $\endgroup$ – Carl Witthoft Jan 24 at 13:34
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This is not an answer, but a step towards one. Perhaps one can find the first user of the term, and find a discussion of the point in question when it is first introduced. A stepping stone in this process is to identify early users of the term.

Heinrich Weber's Lehrbuch der Algebra (Chelsea reprint of 2nd edition of 1899?), volume 2, p.12, defines "Normaltheil" (normal subgroup), and in a footnote gives the synonyms "ausgezeichnete Untergruppe" and "invariente Untergruppe" (distinguished subgroup, invariant subgroup).

The idea of normal subgroups is usually attributed to Galois, who used the term "décomposition propre", according to the discussion in Hans Wussing's 1969 book, published in translation in 1984 (MIT) and 2007 (Dover) as The genesis of the group concept. (See this snippet, for example.)

So maybe "normal" is a translation of "propre".

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