  Large Language Models (LLMs) have demonstrated impressive performance across
a wide range of applications; however, assessing their reasoning capabilities
remains a significant challenge. In this paper, we introduce a framework
grounded in group and symmetry principles, which have played a crucial role in
fields such as physics and mathematics, and offer another way to evaluate their
capabilities. While the proposed framework is general, to showcase the benefits
of employing these properties, we focus on arithmetic reasoning and investigate
the performance of these models on four group properties: closure, identity,
inverse, and associativity. Our findings reveal that LLMs studied in this work
struggle to preserve group properties across different test regimes. In the
closure test, we observe biases towards specific outputs and an abrupt
degradation in their performance from 100% to 0% after a specific sequence
length. They also perform poorly in the identity test, which represents adding
irrelevant information in the context, and show sensitivity when subjected to
inverse test, which examines the robustness of the model with respect to
negation. In addition, we demonstrate that breaking down problems into smaller
steps helps LLMs in the associativity test that we have conducted. To support
these tests we have developed a synthetic dataset which will be released.
