TITLE: Relation between left and right coset representatives of a subgroup
QUESTION [15 upvotes]: Let $G$ be a finite group and $H$ a subgroup. Is it true that
a set of right coset representatives of $H$ is also set of left coset 
representatives of $H$?

REPLY [9 votes]: This answer is mostly to give a nice bibliography for this problem of finding common transversals. I was going to include a proof, but all the proofs were combinatorial and my attempts at making them sound interesting failed.
Exists: yes, All: no
If $H \leq G$ and $K \leq G$ are two subgroups of the same finite index $[G:H]=[G:K]$, then they have a “common transversal”: a set $t_i \in G$ such that $H t_i \neq H t_j$ and $t_i K \neq t_j K$ if $i \neq j$ and such that $G = \cup H t_i = \cup t_i K$. Taking $H=K$ gives an answer to the original question, interpreting "a" as "there exists" rather than "for all". I discuss the publication history of this theorem in the next section.
One can easily check that $\{ (), (2,3), (1,2,3) \}$ is only a one-sided set of coset representatives of $\{(),(1,2)\}$ in the symmetric group $\{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$.
History
The earliest proof is due to Miller (1910) with another similar proof in Chapman (1913). The same proof was used for a stronger result in Scorza (1927). Van der Waerden (1927) proved a combinatorial precursor to the Marriage theorem of Hall (1935) and explicitly cites Miller (1910) as his motivation. A proof is presented in textbook form in Zassenhaus (1937), theorem 3, page 12; I believe this is van der Waerden's proof, but Zassenhaus credits it to Willi Maak. Shü (1941) generalizes Scorza's result to finite index subgroups and gives an example to show the result is not true for a general subgroup. Ore (1958) continues this investigation. Hall (1967) uses the marriage theorem to prove the theorem in the finite index case as theorem 5.1.7 on page 55.  Alonso (1972) proves less, but gives more examples and uses more primitive versions, somewhat similar to Hall (1967). Applegate–Onishi (1977) proves the result for continuous sections of profinite groups.
Bibliography

Miller, G. A.
“On a method due to Galois.”
Quarterly Journal of Mathematics 41, (1910), 382-384.
JFM41.174.1
Chapman, H. W.
“A note on the elementary theory of groups of finite order.”
Messenger of Mathematics (2) 42 (1913), 132-134; correction 43 (1914), 85.
JFM44.168.3
(Correction in next volume)
Scorza, G.
“A proposito di un teorema del Chapman.”
Bollettino della Unione Matemàtica Italiana 6, 1-6 (1927).
JFM53.105.5
van der Waerden, B. L.
“Ein Satz über Klasseneinteilungen von endlichen Mengen.”
Abhandlungen Hamburg 5, 185-187 (1927).
JFM53.171.2
DOI10.1007/BF02952519
Hall, P.
“On representatives of subsets.”
Journ. London Math. Soc. 10, 26-30 (1935).
JFM61.67.1
ZBL10.345.3
DOI:10.1112/jlms/s1-10.37.26
Zassenhaus, Hans.
Lehrbuch der Gruppentheorie.
(Hamburg. Math. Einzelschriften 21) Leipzig, Berlin: B. G. Teubner. VI, 152 S. (1937).
JFM63.58.3
ZBL18.9.1
MR30947
MR91275
Shü, Shien-siu.
“On the common representative system of residue classes of infinite groups.”
J. London Math. Soc. 16, (1941). 101–104.
MR4624
DOI:10.1112/jlms/s1-16.2.101
Ore, Oystein.
“On coset representatives in groups.”
Proc. Amer. Math. Soc. 9 (1958) 665–670.
MR100639
DOI:10.2307/2033229
Hall, Marshall, Jr.
Combinatorial theory.
Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London (1967) x+310 pp.
MR224481
MR840216
Alonso, James.
“Representatives for cosets.”
Amer. Math. Monthly 79 (1972), 886–890.
MR315004
DOI:10.2307/2317669
Appelgate, H.; Onishi, H.
“Coincident pairs of continuous sections in profinite groups.”
Proc. Amer. Math. Soc. 63 (1977), no. 2, 217–220.
MR442099
DOI:10.2307/2041792