Relation between left and right coset representatives of a subgroup

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 $H$?

Not every left transversal is also a right transversal. The Group Properties Wiki has a list of subgroup properties that are stronger than "having [at least one] left transversal that is also a right transversal"; among these is normality as William notes.

However the left coset representatives' multiplicative inverses form a right transversal, because

$$\begin{array}{c l}xH=yH & \iff y^{-1}xH=H \\ & \iff y^{-1}x\in H \\ & \iff (y^{-1}x)^{-1}=x^{-1}y\in H \\ & \iff Hx^{-1}y=H \\ & \iff Hx^{-1}=Hy^{-1}. \end{array}$$

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

No this not true unless $H$ is a normal subgroup. Suppose that $H$ is normal. Then for all $a \in G$, $aHa^{-1} = H$ so $aH = Ha$. Suppose that the set of left and right cosets are the same. Since cosets are partitions of $G$, $aH = Ha$. Hence $aHa^{-1} = H$. Since this holds for all $a \in G$, $H$ is a normal subgroup.

However for any subgroup $H$ of $G$ it is true that the set of left cosets and right cosets are in bijection. Namely $\Phi(aH) = Ha^{-1}$.

Let $R$ and $L$ be the set of right or right and left cosets. Above I showed that $R = L$ if and only $H$ is normal .

Now suppose that $H$ is not normal. So $R \neq L$. Hence there is a right coset $\sigma$ that is not a left coset. Since you assumed that the groups are finite, the size of each left and right cosets are equal. Hence if $\sigma$ is not any left coset, then $\sigma$ must intersect two left cosets $\tau_1$ and $\tau_2$. Let $a_1 \in \sigma \cap \tau_1$ and $a_2 \in \sigma \cap \tau_2$. Now form a set of representatives $K$ for left cosets, where $a_1$ and $a_2$ are choosen representative for $\tau_1$ and $\tau_2$ respectively. Now $K$ is not a set of representative for right cosets since $a_1$ and $a_2$ are representative for the same right coset. So some right coset must have no representative.