= Solution
In <agglomerative hierarchical clustering>, start with eight singleton groups. <Single-linkage clustering> uses the smallest distance between cross-pairs; <complete-linkage clustering> uses the largest. Updating these cluster distances after each merge, one valid order of the tied merges gives
$$
\begin{array}{c|cc|cc}
\text{step}&\text{single cluster}&\text{height}&\text{complete cluster}&\text{height}\\\hline
1&BD&0.4&BD&0.4\\
2&AC&0.6&AC&0.6\\
3&EG&0.6&EG&0.6\\
4&ABCD&0.6&FH&1.0\\
5&EGH&0.9&ABCD&1.4\\
6&EFGH&1.0&EFGH&1.5\\
7&ABCDEFGH&1.9&ABCDEFGH&3.8
\end{array}
$$
The merges at height $0.6$ may be reordered; they give the same requested three-cluster cuts. For example, the single-link distance from $AC$ to $BD$ is $\min(1.2,1.4,0.6,0.8)=0.6$, while its complete-link distance is their maximum $1.4$. After forming $EG$, its single-link distance to $H$ is $\min(0.9,1.2)=0.9$. The complete-link distance between $EG$ and $FH$ is $\max(1.2,0.9,1.5,1.2)=1.5$. Finally the closest cross-pair between $ABCD$ and $EFGH$ is $CG$ at $1.9$, while the farthest is $BF$ at $3.8$.
At three groups the partitions are therefore
$$
\boxed{\text{single linkage: }\{A,B,C,D\},\ \{E,G,H\},\ \{F\};}
$$
$$
\boxed{\text{complete linkage: }\{A,B,C,D\},\ \{E,G\},\ \{F,H\}.}
$$
\b[Two groups, $\{A,B,C,D\}$ and $\{E,F,G,H\}$, are more strongly supported by the merge heights.] For single linkage the three-to-two merge is at $1.0$, close to the preceding height $0.9$, whereas the two-to-one merge is delayed until $1.9$. For complete linkage the three-to-two merge at $1.5$ is close to $1.4$, whereas the final merge jumps to $3.8$. Both methods agree on the two-group partition but split its second group differently when forced to return three groups. A three-group choice is possible if required substantively, but is not particularly compelling from these dissimilarities alone.
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