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Let $G$ be a group. Suppose there exist a finite sequence of elements $a_1, \cdots, a_n$ and a finite sequence of subgroups $H_1, \cdots, H_n$ such that $G = \bigcup_{i=1}^n a_iH_i$ is a disjoint union. If $(G: H_1) = \cdots = (G : H_n) \lt \infty$, then $H_1= \cdots = H_n?$

I came up with this problem when trying to solve this question. Explicit construction of Haar measure on a profinite group

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JHW
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Let $G = \{ 1,a,b,c,ab,ac,abc,bc\}$ be elementary abelian of order $8$. Then $$ G = \{1,a\} \cup \{c,ac\} \cup \{b,bc\} \cup \{ab,abc\}$$ which is the union of two cosets of the subgroup $\{1,a\}$ and two of the subgroup $\{1,c\}$.

Derek Holt
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